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Separable extension

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In <a href="Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a>, a branch of <a href="Algebra" title="Algebra">algebra</a>, an <a href="Algebraic_field_extension" class="mw-redirect" title="Algebraic field extension">algebraic field extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E/F}">

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<annotation encoding="application/x-tex">{\displaystyle E/F}</annotation>

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</math><img src="./f86227a53670608bf39758bc69a4529774756b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.679ex; height:2.843ex;" alt="{\displaystyle E/F}" loading="lazy"> is called a separable extension if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in E}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \in E}</annotation>

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</math><img src="./cd076032ef2ac2d9ab57f247fc5a7f172bf9462c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.104ex; height:2.176ex;" alt="{\displaystyle \alpha \in E}" loading="lazy">, the <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> over F is a <a href="Separable_polynomial" title="Separable polynomial">separable polynomial</a> (i.e., its <a href="Formal_derivative" title="Formal derivative">formal derivative</a> is not the zero <a href="Polynomial" title="Polynomial">polynomial</a>, or equivalently it has no repeated <a href="Zero_of_a_function" title="Zero of a function">roots</a> in any extension field).<a href="#cite_note-Isaacs281-1">[1]</a> There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.

Every algebraic extension of a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="Characteristic_(algebra)#Case_of_fields" title="Characteristic (algebra)">characteristic</a> zero is separable, and every algebraic extension of a <a href="Finite_field" title="Finite field">finite field</a> is separable.<a href="#cite_note-Isaacs18.11p281-2">[2]</a>

It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the <a href="Fundamental_theorem_of_Galois_theory" title="Fundamental theorem of Galois theory">fundamental theorem of Galois theory</a> is a theorem about <a href="Normal_extension" title="Normal extension">normal extensions</a>, which remains true in non-zero characteristic only if the extensions are also assumed to be separable.<a href="#cite_note-3">[3]</a>

The opposite concept, a <a href="Purely_inseparable_extension" title="Purely inseparable extension">purely inseparable extension</a>, also occurs naturally, as every algebraic extension may be decomposed uniquely as a purely inseparable extension of a separable extension. An algebraic extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E/F}">

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<annotation encoding="application/x-tex">{\displaystyle E/F}</annotation>

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</math><img src="./f86227a53670608bf39758bc69a4529774756b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.679ex; height:2.843ex;" alt="{\displaystyle E/F}" loading="lazy"> of fields of non-zero characteristic p is a purely inseparable extension if and only if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in E\setminus F}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \in E\setminus F}</annotation>

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</math><img src="./208f46fbc31cc0f8a48116c83662a89255286eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.039ex; height:2.843ex;" alt="{\displaystyle \alpha \in E\setminus F}" loading="lazy">, the minimal polynomial of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> over F is not a separable polynomial, or, equivalently, for every element x of E, there is a positive <a href="Integer" title="Integer">integer</a> k such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{p^{k}}\in F}">

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<annotation encoding="application/x-tex">{\displaystyle x^{p^{k}}\in F}</annotation>

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</math><img src="./dba2257f346d7facfac16e8709834fdfedc40e2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.83ex; height:3.009ex;" alt="{\displaystyle x^{p^{k}}\in F}" loading="lazy">.<a href="#cite_note-Isaacs298-4">[4]</a>

The simplest nontrivial example of a (purely) inseparable extension is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\mathbb {F} _{p}(x)\supseteq F=\mathbb {F} _{p}(x^{p})}">

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<annotation encoding="application/x-tex">{\displaystyle E=\mathbb {F} _{p}(x)\supseteq F=\mathbb {F} _{p}(x^{p})}</annotation>

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</math><img src="./5b4b3847a19a548101e2a14d2b2edc23d07d7cbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.108ex; height:3.009ex;" alt="{\displaystyle E=\mathbb {F} _{p}(x)\supseteq F=\mathbb {F} _{p}(x^{p})}" loading="lazy">, fields of <a href="Rational_function" title="Rational function">rational functions</a> in the indeterminate x with coefficients in the <a href="Finite_field" title="Finite field">finite field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}=\mathbb {Z} /(p)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}=\mathbb {Z} /(p)}</annotation>

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</math><img src="./6193c6b025c5cb2b2f1651f9ae73e7c49e6d19f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.269ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{p}=\mathbb {Z} /(p)}" loading="lazy">. The element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E}">

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<annotation encoding="application/x-tex">{\displaystyle x\in E}</annotation>

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</math><img src="./30b1971b01bc31d5b816f03cc7e1d9215d6c2ad8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.946ex; height:2.176ex;" alt="{\displaystyle x\in E}" loading="lazy"> has minimal polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)=X^{p}-x^{p}\in F[X]}">

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<annotation encoding="application/x-tex">{\displaystyle f(X)=X^{p}-x^{p}\in F[X]}</annotation>

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</math><img src="./aa90729f83deabe4f595ffffe336899de9bdd119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.307ex; height:2.843ex;" alt="{\displaystyle f(X)=X^{p}-x^{p}\in F[X]}" loading="lazy">, having <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(X)=0}">

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<annotation encoding="application/x-tex">{\displaystyle f'(X)=0}</annotation>

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</math><img src="./f22834786924c67572695ed818297ad5709c80ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.055ex; height:3.009ex;" alt="{\displaystyle f'(X)=0}" loading="lazy"> and a p-fold multiple root, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)=(X-x)^{p}\in E[X]}">

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<annotation encoding="application/x-tex">{\displaystyle f(X)=(X-x)^{p}\in E[X]}</annotation>

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</math><img src="./1e19287afa379f76b1725ea2d60ed72a08b3aad7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.075ex; height:2.843ex;" alt="{\displaystyle f(X)=(X-x)^{p}\in E[X]}" loading="lazy">. This is a <a href="Simple_field_extension" class="mw-redirect" title="Simple field extension">simple</a> algebraic extension of degree p, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=F[x]}">

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<annotation encoding="application/x-tex">{\displaystyle E=F[x]}</annotation>

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</math><img src="./c3e2ffe95fea7b2f3be3d049bd921e4702513976.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.238ex; height:2.843ex;" alt="{\displaystyle E=F[x]}" loading="lazy">, but it is not a normal extension since the <a href="Galois_group" title="Galois group">Galois group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Gal}}(E/F)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Gal}}(E/F)}</annotation>

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</math><img src="./826059e8b75736915a8765a64de3a3bea1fe7e30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.122ex; height:2.843ex;" alt="{\displaystyle {\text{Gal}}(E/F)}" loading="lazy"> is <a href="Trivial_group" title="Trivial group">trivial</a>.

<div class="mw-heading mw-heading2"><h2 id="Informal_discussion">Informal discussion</h2></div>

An arbitrary polynomial f with coefficients in some field F is said to have distinct roots or to be <a href="Square-free_polynomial" title="Square-free polynomial">square-free</a> if it has deg f roots in some <a href="Extension_field" class="mw-redirect" title="Extension field">extension field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

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<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

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</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy">. For instance, the polynomial g(X) = X 2 − 1 has precisely deg g = 2 roots in the <a href="Complex_plane" title="Complex plane">complex plane</a>; namely 1 and −1, and hence does have distinct roots. On the other hand, the polynomial h(X) = (X − 2)2, which is the square of a non-constant polynomial does not have distinct roots, as its degree is two, and 2 is its only root.

Every polynomial may be factored in linear factors over an <a href="Algebraic_closure" title="Algebraic closure">algebraic closure</a> of the field of its coefficients. Therefore, the polynomial does not have distinct roots if and only if it is divisible by the square of a polynomial of positive degree. This is the case if and only if the <a href="Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">greatest common divisor</a> of the polynomial and its <a href="Formal_derivative" title="Formal derivative">derivative</a> is not a constant. Thus for testing if a polynomial is square-free, it is not necessary to consider explicitly any field extension nor to compute the roots.

In this context, the case of irreducible polynomials requires some care. A priori, it may seem that being divisible by a square is impossible for an <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a>, which has no non-constant divisor except itself. However, irreducibility depends on the ambient field, and a polynomial may be irreducible over F and reducible over some extension of F. Similarly, divisibility by a square depends on the ambient field. If an irreducible polynomial f over F is divisible by a square over some field extension, then (by the discussion above) the greatest common divisor of f and its derivative f′ is not constant. Note that the coefficients of f′ belong to the same field as those of f, and the greatest common divisor of two polynomials is independent of the ambient field, so the greatest common divisor of f and f′ has coefficients in F. Since f is irreducible in F, this greatest common divisor is necessarily f itself. Because the degree of f′ is strictly less than the degree of f, it follows that the derivative of f is zero, which implies that the <a href="Characteristic_of_a_field" class="mw-redirect" title="Characteristic of a field">characteristic</a> of the field is a prime number p, and f may be written

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{i=0}^{k}a_{i}x^{pi}.}</annotation>

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</math><img src="./97de4a6709e63c7f0b0b3ccd0f97c3ad3113ff53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.891ex; height:7.343ex;" alt="{\displaystyle f(x)=\sum _{i=0}^{k}a_{i}x^{pi}.}" loading="lazy"></dd></dl>

A polynomial such as this one, whose formal derivative is zero, is said to be inseparable. Polynomials that are not inseparable are said to be separable. A separable extension is an extension that may be generated by separable elements, that is elements whose minimal polynomials are separable.

<div class="mw-heading mw-heading2"><h2 id="Separable_and_inseparable_polynomials">Separable and inseparable polynomials</h2></div>

An <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a> f in F[X] is <a href="Separable_polynomial" title="Separable polynomial">separable</a> if and only if it has distinct roots in any <a href="Field_extension" title="Field extension">extension</a> of F. That is, if it is the product of distinct linear factors X - a in some <a href="Algebraically_closed_field" title="Algebraically closed field">algebraic closure</a> of F.<a href="#cite_note-5">[5]</a>

Let f in F[X] be an irreducible polynomial and f ' its <a href="Formal_derivative" title="Formal derivative">formal derivative</a>. Then the following are equivalent conditions for the irreducible polynomial f to be separable:

<ul><li>If E is an extension of F in which f is a product of linear factors then no square of these factors divides f in E[X] (that is f is <a href="Square-free_polynomial" title="Square-free polynomial">square-free</a> over E).<a href="#cite_note-IsaacsLem18.7-6">[6]</a></li>

<li>There exists an extension E of F such that f has deg(f) pairwise distinct roots in E.<a href="#cite_note-IsaacsLem18.7-6">[6]</a></li>

<li>The constant 1 is a <a href="Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">polynomial greatest common divisor</a> of f and f '.<a href="#cite_note-7">[7]</a></li>

<li>The formal derivative f ' of f is not the zero polynomial.<a href="#cite_note-8">[8]</a></li>

<li>Either the characteristic of F is zero, or the characteristic is p, and f is not of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i=0}^{k}a_{i}X^{pi}.}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i=0}^{k}a_{i}X^{pi}.}</annotation>

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</math><img src="./d27acbd8ea016672727bfba200decf2f5a2bcec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.041ex; height:3.509ex;" alt="{\displaystyle \textstyle \sum _{i=0}^{k}a_{i}X^{pi}.}" loading="lazy"></li></ul>

Since the formal derivative of a positive degree polynomial can be zero only if the field has prime characteristic, for an irreducible polynomial to not be separable, its coefficients must lie in a field of prime characteristic. More generally, an irreducible (non-zero) polynomial f in F[X] is not separable, if and only if the characteristic of F is a (non-zero) prime number p, and f(X)=g(Xp) for some irreducible polynomial g in F[X].<a href="#cite_note-9">[9]</a> By repeated application of this property, it follows that in fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)=g(X^{p^{n}})}">

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<annotation encoding="application/x-tex">{\displaystyle f(X)=g(X^{p^{n}})}</annotation>

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</math><img src="./c31549203fd7181c2310998ab592ff9d58efda3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.113ex; height:3.176ex;" alt="{\displaystyle f(X)=g(X^{p^{n}})}" loading="lazy"> for a non-negative integer n and some separable irreducible polynomial g in F[X] (where F is assumed to have prime characteristic p).<a href="#cite_note-10">[10]</a>

If the <a href="Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x^{p}}">

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{p}}</annotation>

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</math><img src="./8f72ef9f7692b1b55c635323f2287d96b1d8cec7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:2.343ex;" alt="{\displaystyle x\mapsto x^{p}}" loading="lazy"> of F is not surjective, there is an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in F}">

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<annotation encoding="application/x-tex">{\displaystyle a\in F}</annotation>

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</math><img src="./130d30b75a5437ad01787d25462043ac3a9aee3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.811ex; height:2.176ex;" alt="{\displaystyle a\in F}" loading="lazy"> that is not a pth power of an element of F. In this case, the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{p}-a}">

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<annotation encoding="application/x-tex">{\displaystyle X^{p}-a}</annotation>

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</math><img src="./f8c89727242046a8131f11cd9925bb3b32903a39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.126ex; height:2.509ex;" alt="{\displaystyle X^{p}-a}" loading="lazy"> is irreducible and inseparable. Conversely, if there exists an inseparable irreducible (non-zero) polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f(X)=\sum a_{i}X^{ip}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle f(X)=\sum a_{i}X^{ip}}</annotation>

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</math><img src="./a4e38cd33416bb9e440e3326c5d54f8d42fd05f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.66ex; height:3.009ex;" alt="{\displaystyle \textstyle f(X)=\sum a_{i}X^{ip}}" loading="lazy"> in F[X], then the <a href="Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a> of F cannot be an <a href="Automorphism" title="Automorphism">automorphism</a>, since, otherwise, we would have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=b_{i}^{p}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{i}=b_{i}^{p}}</annotation>

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</math><img src="./7e376ee3fb9b1a34193e83e6cef515e38247950a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.185ex; height:3.176ex;" alt="{\displaystyle a_{i}=b_{i}^{p}}" loading="lazy"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{i}}</annotation>

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</math><img src="./40a8c2db2990a53c683e75961826167c5adac7c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.797ex; height:2.509ex;" alt="{\displaystyle b_{i}}" loading="lazy">, and the polynomial f would factor as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum a_{i}X^{ip}=\left(\sum b_{i}X^{i}\right)^{p}.}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum a_{i}X^{ip}=\left(\sum b_{i}X^{i}\right)^{p}.}</annotation>

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</math><img src="./21b294bc0f5036d21726e55b84856f71e96f2664.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.863ex; height:3.343ex;" alt="{\displaystyle \textstyle \sum a_{i}X^{ip}=\left(\sum b_{i}X^{i}\right)^{p}.}" loading="lazy"><a href="#cite_note-11">[11]</a>

If K is a finite field of prime characteristic p, and if X is an <a href="Indeterminate_(variable)" class="mw-redirect" title="Indeterminate (variable)">indeterminate</a>, then the <a href="Field_of_rational_functions" class="mw-redirect" title="Field of rational functions">field of rational functions</a> over K, K(X), is necessarily <a href="Imperfect_field" class="mw-redirect" title="Imperfect field">imperfect</a>, and the polynomial f(Y)=Yp−X is inseparable (its formal derivative in Y is 0).<a href="#cite_note-Isaacs281-1">[1]</a> More generally, if F is any field of (non-zero) prime characteristic for which the <a href="Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a> is not an automorphism, F possesses an inseparable algebraic extension.<a href="#cite_note-Isaacs299-12">[12]</a>

A field F is <a href="Perfect_field" title="Perfect field">perfect</a> if and only if all irreducible polynomials are separable. It follows that F is perfect if and only if either F has characteristic zero, or F has (non-zero) prime characteristic p and the <a href="Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a> of F is an automorphism. This includes every finite field.

<div class="mw-heading mw-heading2"><h2 id="Separable_elements_and_separable_extensions">Separable elements and separable extensions</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

</semantics>

</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy"> be a field extension. An element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in E}</annotation>

</semantics>

</math><img src="./cd076032ef2ac2d9ab57f247fc5a7f172bf9462c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.104ex; height:2.176ex;" alt="{\displaystyle \alpha \in E}" loading="lazy"> is separable over F if it is algebraic over F, and its <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> is separable (the minimal polynomial of an element is necessarily irreducible).

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta \in E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta \in E}</annotation>

</semantics>

</math><img src="./e05e70b301b09312526616b214013d6af1653991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.47ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta \in E}" loading="lazy"> are separable over F, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha +\beta }</annotation>

</semantics>

</math><img src="./99b80a3fdecb9cf75091789bb4335a1bb3561b08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.66ex; height:2.509ex;" alt="{\displaystyle \alpha +\beta }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \beta }</annotation>

</semantics>

</math><img src="./2efb0e5523f52275f3193b0dfd9a92ad5b76830c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.82ex; height:2.509ex;" alt="{\displaystyle \alpha \beta }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\alpha }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/\alpha }</annotation>

</semantics>

</math><img src="./9d4038198668caf6b8f9f59c2ff86e7ebb7e7d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.813ex; height:2.843ex;" alt="{\displaystyle 1/\alpha }" loading="lazy"> are separable over F.

Thus the set of all elements in E separable over F forms a subfield of E, called the separable closure of F in E.<a href="#cite_note-13">[13]</a>

The separable closure of F in an <a href="Algebraic_closure" title="Algebraic closure">algebraic closure</a> of F is simply called the <a href="Separable_closure" class="mw-redirect" title="Separable closure">separable closure</a> of F. Like the algebraic closure, it is unique up to an isomorphism, and in general, this isomorphism is not unique.

A field extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

</semantics>

</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy"> is separable, if E is the separable closure of F in E. This is the case if and only if E is generated over F by separable elements.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq L\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq L\supseteq F}</annotation>

</semantics>

</math><img src="./9275514242d7c9198c203f397ff39fdb239ba547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.296ex; height:2.343ex;" alt="{\displaystyle E\supseteq L\supseteq F}" loading="lazy"> are field extensions, then E is separable over F if and only if E is separable over L and L is separable over F.<a href="#cite_note-14">[14]</a>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

</semantics>

</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy"> is a <a href="Finite_extension" class="mw-redirect" title="Finite extension">finite extension</a> (that is E is a F-<a href="Vector_space" title="Vector space">vector space</a> of finite <a href="Dimension_(vector_space)" title="Dimension (vector space)">dimension</a>), then the following are equivalent.

<ol><li>E is separable over F.</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=F(a_{1},\ldots ,a_{r})}</annotation>

</semantics>

</math><img src="./2a5e10c85998550f7803d4c963a4d436133b4b09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.09ex; height:2.843ex;" alt="{\displaystyle E=F(a_{1},\ldots ,a_{r})}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{r}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{r}}</annotation>

</semantics>

</math><img src="./f0b2813be0d50581d8ee1e046756858d2b360c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.666ex; height:2.009ex;" alt="{\displaystyle a_{1},\ldots ,a_{r}}" loading="lazy"> are separable elements of E.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=F(a)}</annotation>

</semantics>

</math><img src="./8259e6ab9855cb793a29092644379c0e082a515f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle E=F(a)}" loading="lazy"> where a is a separable element of E.</li>

<li>If K is an algebraic closure of F, then there are exactly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [E:F]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [E:F]}</annotation>

</semantics>

</math><img src="./9490bd8939b0fb2663f3f26dac59badef0184170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.747ex; height:2.843ex;" alt="{\displaystyle [E:F]}" loading="lazy"> <a href="Field_homomorphism" class="mw-redirect" title="Field homomorphism">field homomorphisms</a> of E into K that fix F.</li>

<li>For any normal extension K of F that contains E, then there are exactly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [E:F]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [E:F]}</annotation>

</semantics>

</math><img src="./9490bd8939b0fb2663f3f26dac59badef0184170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.747ex; height:2.843ex;" alt="{\displaystyle [E:F]}" loading="lazy"> field homomorphisms of E into K that fix F.</li></ol>

The equivalence of 3. and 1. is known as the <a href="Primitive_element_theorem" title="Primitive element theorem">primitive element theorem</a> or Artin's theorem on primitive elements.

Properties 4. and 5. are the basis of <a href="Galois_theory" title="Galois theory">Galois theory</a>, and, in particular, of the <a href="Fundamental_theorem_of_Galois_theory" title="Fundamental theorem of Galois theory">fundamental theorem of Galois theory</a>.

<div class="mw-heading mw-heading2"><h2 id="Separable_extensions_within_algebraic_extensions">Separable extensions within algebraic extensions</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

</semantics>

</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy"> be an algebraic extension of fields of characteristic p. The separable closure of F in E is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{\alpha \in E\mid \alpha {\text{ is separable over }}F\}.}">

<mtext> is separable over </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\{\alpha \in E\mid \alpha {\text{ is separable over }}F\}.}</annotation>

</semantics>

</math><img src="./98fedf0b5e4f38d8bb1d0a87c4dd9c1b3ef339d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.508ex; height:2.843ex;" alt="{\displaystyle S=\{\alpha \in E\mid \alpha {\text{ is separable over }}F\}.}" loading="lazy"> For every element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E\setminus S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in E\setminus S}</annotation>

</semantics>

</math><img src="./731a116a9205ac45fa827bb891ada1950859c0f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.64ex; height:2.843ex;" alt="{\displaystyle x\in E\setminus S}" loading="lazy"> there exists a positive integer k such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{p^{k}}\in S,}">

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{p^{k}}\in S,}</annotation>

</semantics>

</math><img src="./15b4ecf511352c86d084c7d3341f6a9952af3794.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.235ex; height:3.343ex;" alt="{\displaystyle x^{p^{k}}\in S,}" loading="lazy"> and thus E is a <a href="Purely_inseparable_extension" title="Purely inseparable extension">purely inseparable extension</a> of S. It follows that S is the unique intermediate field that is separable over F and over which E is purely inseparable.<a href="#cite_note-15">[15]</a>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

</semantics>

</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy"> is a <a href="Finite_extension" class="mw-redirect" title="Finite extension">finite extension</a>, its <a href="Degree_of_a_field_extension" title="Degree of a field extension">degree</a> [E : F] is the product of the degrees [S : F] and [E : S]. The former, often denoted [E : F]sep, is referred to as the separable part of [E : F], or as the <style data-mw-deduplicate="TemplateStyles:r1238216509">

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.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}

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</style>separable degree of E/F; the latter is referred to as the inseparable part of the degree or the inseparable degree.<a href="#cite_note-Isaacs302-16">[16]</a> The inseparable degree is 1 in characteristic zero and a power of p in characteristic p > 0.<a href="#cite_note-17">[17]</a>

On the other hand, an arbitrary algebraic extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

</semantics>

</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy"> may not possess an intermediate extension K that is purely inseparable over F and over which E is separable. However, such an intermediate extension may exist if, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

</semantics>

</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy"> is a finite degree normal extension (in this case, K is the fixed field of the Galois group of E over F). Suppose that such an intermediate extension does exist, and [E : F] is finite, then [S : F] = [E : K], where S is the separable closure of F in E.<a href="#cite_note-18">[18]</a> The known proofs of this equality use the fact that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\supseteq F}</annotation>

</semantics>

</math><img src="./87423fa393c0801fa495ab788474ccce1ea0cf73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.905ex; height:2.343ex;" alt="{\displaystyle K\supseteq F}" loading="lazy"> is a purely inseparable extension, and if f is a separable irreducible polynomial in F[X], then f remains irreducible in K[X]<a href="#cite_note-19">[19]</a>). This equality implies that, if [E : F] is finite, and U is an intermediate field between F and E, then [E : F]sep = [E : U]sep⋅[U : F]sep.<a href="#cite_note-20">[20]</a>

The separable closure Fsep of a field F is the separable closure of F in an <a href="Algebraic_closure" title="Algebraic closure">algebraic closure</a> of F. It is the maximal <a href="Galois_extension" title="Galois extension">Galois extension</a> of F. By definition, F is <a href="Perfect_field" title="Perfect field">perfect</a> if and only if its separable and algebraic closures coincide.

<div class="mw-heading mw-heading2"><h2 id="Separability_of_transcendental_extensions">Separability of transcendental extensions</h2></div>

Separability problems may arise when dealing with <a href="Transcendental_extension" title="Transcendental extension">transcendental extensions</a>. This is typically the case for <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> over a field of prime characteristic, where the <a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field of an algebraic variety</a> has a <a href="Transcendence_degree" class="mw-redirect" title="Transcendence degree">transcendence degree</a> over the ground field that is equal to the <a href="Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">dimension</a> of the variety.

For defining the separability of a transcendental extension, it is natural to use the fact that every field extension is an algebraic extension of a <a href="Purely_transcendental_extension" class="mw-redirect" title="Purely transcendental extension">purely transcendental extension</a>. This leads to the following definition.

A separating transcendence basis of an extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

</semantics>

</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy"> is a <a href="Transcendence_basis" class="mw-redirect" title="Transcendence basis">transcendence basis</a> T of E such that E is a separable algebraic extension of F(T). A <a href="Finitely_generated_field_extension" class="mw-redirect" title="Finitely generated field extension">finitely generated field extension</a> is separable if and only it has a separating transcendence basis; an extension that is not finitely generated is called separable if every finitely generated subextension has a separating transcendence basis.<a href="#cite_note-FJ38-21">[21]</a>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\supseteq F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\supseteq F}</annotation>

</semantics>

</math><img src="./b033b5b0999de90ad047151b31d4ce3be38b9ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle E\supseteq F}" loading="lazy"> be a field extension of <a href="Characteristic_exponent_of_a_field" class="mw-redirect" title="Characteristic exponent of a field">characteristic exponent</a> p (that is p = 1 in characteristic zero and, otherwise, p is the characteristic). The following properties are equivalent:

<ul><li>E is a separable extension of F,</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E^{p}}</annotation>

</semantics>

</math><img src="./9dbff0c5189ba40e3492adff94891171e73af8fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.853ex; height:2.343ex;" alt="{\displaystyle E^{p}}" loading="lazy"> and F are <a href="Linearly_disjoint" class="mw-redirect" title="Linearly disjoint">linearly disjoint</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{p},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{p},}</annotation>

</semantics>

</math><img src="./a926ce6878ad89a30a5e079023c2269600411482.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.521ex; height:2.676ex;" alt="{\displaystyle F^{p},}" loading="lazy"></li>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{1/p}\otimes _{F}E}</annotation>

</semantics>

</math><img src="./6310a014414420b6ad179612d3e66cbe6d044544.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.597ex; height:3.176ex;" alt="{\displaystyle F^{1/p}\otimes _{F}E}" loading="lazy"> is <a href="Reduced_ring" title="Reduced ring">reduced</a>,</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L\otimes _{F}E}</annotation>

</semantics>

</math><img src="./20578a8e180fda4d72db95a030411b760bd40d3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.662ex; height:2.509ex;" alt="{\displaystyle L\otimes _{F}E}" loading="lazy"> is reduced for every field extension L of E,</li></ul>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes _{F}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \otimes _{F}}</annotation>

</semantics>

</math><img src="./63fc69d07e96cedd8fa958e5bd75e7f1229dc55b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.271ex; height:2.343ex;" alt="{\displaystyle \otimes _{F}}" loading="lazy"> denotes the <a href="Tensor_product_of_fields" title="Tensor product of fields">tensor product of fields</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{p}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{p}}</annotation>

</semantics>

</math><img src="./eb22f014af2031193d67a74c919d10c678c6bcaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.874ex; height:2.343ex;" alt="{\displaystyle F^{p}}" loading="lazy"> is the field of the pth powers of the elements of F (for any field F), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{1/p}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{1/p}}</annotation>

</semantics>

</math><img src="./04a02880a5b5af86d751429764911eb58244bda0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.518ex; height:2.843ex;" alt="{\displaystyle F^{1/p}}" loading="lazy"> is the field obtained by <a href="Adjunction_(field_theory)" class="mw-redirect" title="Adjunction (field theory)">adjoining</a> to F the pth root of all its elements (see <a href="Separable_algebra" title="Separable algebra">Separable algebra</a> for details).

<div class="mw-heading mw-heading2"><h2 id="Differential_criteria">Differential criteria</h2></div>

Separability can be studied with the aid of <a href="K%C3%A4hler_differential" title="Kähler differential">derivations</a>. Let E be a <a href="Finitely_generated_field_extension" class="mw-redirect" title="Finitely generated field extension">finitely generated field extension</a> of a field F. Denoting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Der} _{F}(E,E)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Der} _{F}(E,E)}</annotation>

</semantics>

</math><img src="./dc9d9c6f263f11a8a668cd750e8a838a1b2126de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.577ex; height:2.843ex;" alt="{\displaystyle \operatorname {Der} _{F}(E,E)}" loading="lazy"> the E-vector space of the F-linear derivations of E, one has

<msub>

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</msub>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim _{E}\operatorname {Der} _{F}(E,E)\geq \operatorname {tr.deg} _{F}E,}</annotation>

</semantics>

</math><img src="./cd617bb9f54fb4d3388568ee6bd5f10dc909fdd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.036ex; height:2.843ex;" alt="{\displaystyle \dim _{E}\operatorname {Der} _{F}(E,E)\geq \operatorname {tr.deg} _{F}E,}" loading="lazy"></dd></dl>

and the equality holds if and only if E is separable over F (here "tr.deg" denotes the <a href="Transcendence_degree" class="mw-redirect" title="Transcendence degree">transcendence degree</a>).

In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E/F}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E/F}</annotation>

</semantics>

</math><img src="./f86227a53670608bf39758bc69a4529774756b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.679ex; height:2.843ex;" alt="{\displaystyle E/F}" loading="lazy"> is an algebraic extension, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Der} _{F}(E,E)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Der} _{F}(E,E)=0}</annotation>

</semantics>

</math><img src="./0eea084f792a06fad418a199019422332e5c3cb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.838ex; height:2.843ex;" alt="{\displaystyle \operatorname {Der} _{F}(E,E)=0}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E/F}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E/F}</annotation>

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</math><img src="./f86227a53670608bf39758bc69a4529774756b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.679ex; height:2.843ex;" alt="{\displaystyle E/F}" loading="lazy"> is separable.<a href="#cite_note-FJ49-22">[22]</a>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{1},\ldots ,D_{m}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{1},\ldots ,D_{m}}</annotation>

</semantics>

</math><img src="./f23496a42e529a9cac854baaf480a0add538a0e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.756ex; height:2.509ex;" alt="{\displaystyle D_{1},\ldots ,D_{m}}" loading="lazy"> be a basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Der} _{F}(E,E)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Der} _{F}(E,E)}</annotation>

</semantics>

</math><img src="./dc9d9c6f263f11a8a668cd750e8a838a1b2126de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.577ex; height:2.843ex;" alt="{\displaystyle \operatorname {Der} _{F}(E,E)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{m}\in E}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{m}\in E}</annotation>

</semantics>

</math><img src="./426ba774c7224fcc128dbfcda72db13d66d8f3ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.983ex; height:2.509ex;" alt="{\displaystyle a_{1},\ldots ,a_{m}\in E}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is separable algebraic over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(a_{1},\ldots ,a_{m})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(a_{1},\ldots ,a_{m})}</annotation>

</semantics>

</math><img src="./3de8885743399220da308e0bd3eeab5bbd3ec062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.917ex; height:2.843ex;" alt="{\displaystyle F(a_{1},\ldots ,a_{m})}" loading="lazy"> if and only if the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}(a_{j})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{i}(a_{j})}</annotation>

</semantics>

</math><img src="./c24d19707c42b8143c7bfb4a0ac28e8072287267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.673ex; height:3.009ex;" alt="{\displaystyle D_{i}(a_{j})}" loading="lazy"> is invertible. In particular, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=\operatorname {tr.deg} _{F}E}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=\operatorname {tr.deg} _{F}E}</annotation>

</semantics>

</math><img src="./ee3241f51746f41f3f3cfdf7f5e85b85337a6f9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.102ex; height:2.676ex;" alt="{\displaystyle m=\operatorname {tr.deg} _{F}E}" loading="lazy">, this matrix is invertible if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{1},\ldots ,a_{m}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{a_{1},\ldots ,a_{m}\}}</annotation>

</semantics>

</math><img src="./c85cc3897cf0a4f5245dbd4dc189e245a7ec69c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.692ex; height:2.843ex;" alt="{\displaystyle \{a_{1},\ldots ,a_{m}\}}" loading="lazy"> is a separating transcendence basis.

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">

<li id="cite_note-Isaacs281-1">^ <a href="#cite_ref-Isaacs281_1-0">a</a> <a href="#cite_ref-Isaacs281_1-1">b</a> Isaacs, p. 281

</li>

<li id="cite_note-Isaacs18.11p281-2"><a href="#cite_ref-Isaacs18.11p281_2-0">^</a> Isaacs, Theorem 18.11, p. 281

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Isaacs, Theorem 18.13, p. 282

</li>

<li id="cite_note-Isaacs298-4"><a href="#cite_ref-Isaacs298_4-0">^</a> Isaacs, p. 298

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Isaacs, p. 280

</li>

<li id="cite_note-IsaacsLem18.7-6">^ <a href="#cite_ref-IsaacsLem18.7_6-0">a</a> <a href="#cite_ref-IsaacsLem18.7_6-1">b</a> Isaacs, Lemma 18.7, p. 280

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Isaacs, Theorem 19.4, p. 295

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Isaacs, Corollary 19.5, p. 296

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> Isaacs, Corollary 19.6, p. 296

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Isaacs, Corollary 19.9, p. 298

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> Isaacs, Theorem 19.7, p. 297

</li>

<li id="cite_note-Isaacs299-12"><a href="#cite_ref-Isaacs299_12-0">^</a> Isaacs, p. 299

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> Isaacs, Lemma 19.15, p. 300

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> Isaacs, Corollary 18.12, p. 281 and Corollary 19.17, p. 301

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> Isaacs, Theorem 19.14, p. 300

</li>

<li id="cite_note-Isaacs302-16"><a href="#cite_ref-Isaacs302_16-0">^</a> Isaacs, p. 302

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, Corollary V.6.2

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> Isaacs, Theorem 19.19, p. 302

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> Isaacs, Lemma 19.20, p. 302

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> Isaacs, Corollary 19.21, p. 303

</li>

<li id="cite_note-FJ38-21"><a href="#cite_ref-FJ38_21-0">^</a> Fried & Jarden (2008) p.38

</li>

<li id="cite_note-FJ49-22"><a href="#cite_ref-FJ49_22-0">^</a> Fried & Jarden (2008) p.49

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>Borel, A. Linear algebraic groups, 2nd ed.</li>

<li>P.M. Cohn (2003). Basic algebra</li>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFFriedJarden2008" class="citation book cs1">Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd ed.). <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-77269-9</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1145.12001">1145.12001</a>.</cite></li>

<li><cite id="CITEREFI._Martin_Isaacs1993" class="citation book cs1"><a href="Martin_Isaacs" class="mw-redirect" title="Martin Isaacs">I. Martin Isaacs</a> (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-19002-2</bdi>.</cite></li>

<li><cite id="CITEREFKaplansky1972" class="citation book cs1"><a href="Irving_Kaplansky" title="Irving Kaplansky">Kaplansky, Irving</a> (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University of Chicago Press. pp. 55–59. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-226-42451-0</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1001.16500">1001.16500</a>.</cite></li>

<li><cite id="CITEREFLang2002" class="citation cs2"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (2002), <a href="Algebra_(Lang)" title="Algebra (Lang)">Algebra</a>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 211 (Revised third ed.), New York: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95385-4</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556">1878556</a></cite></li>

<li>M. Nagata (1985). Commutative field theory: new edition, Shokabo. (Japanese) <a rel="nofollow" class="external autonumber" href="http://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1309-8.htm">[1]</a></li>

<li><cite id="CITEREFSilverman1993" class="citation book cs1">Silverman, Joseph (1993). The Arithmetic of Elliptic Curves. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-96203-4</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=separable_extension_of_a_field_k">"separable extension of a field k"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-03-18" href="https://en.wikipedia.org/wiki/?title=Separable_extension&oldid=1281088265">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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