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Algebraic function field

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In <a href="Mathematics" title="Mathematics">mathematics</a>, an algebraic function field (often abbreviated as function field) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> variables over a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is a finitely generated <a href="Field_extension" title="Field extension">field extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/k}">

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

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</math><img src="./dea0107b51200e74c3cd319d023acdb194ebd506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.44ex; height:2.843ex;" alt="{\displaystyle K/k}" loading="lazy"> which has <a href="Transcendence_degree" class="mw-redirect" title="Transcendence degree">transcendence degree</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.<a href="#cite_note-1">[1]</a> Equivalently, an algebraic function field of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> variables over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> may be defined as a <a href="Finite_field_extension" class="mw-redirect" title="Finite field extension">finite field extension</a> of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=k(x_{1},\dots ,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle K=k(x_{1},\dots ,x_{n})}</annotation>

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</math><img src="./3254d0e522cbc772fbb18bcd94fcc2b188117d49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.295ex; height:2.843ex;" alt="{\displaystyle K=k(x_{1},\dots ,x_{n})}" loading="lazy"> of <a href="Rational_functions" class="mw-redirect" title="Rational functions">rational functions</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> variables over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.

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

As an example, in the <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x,y]}">

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<annotation encoding="application/x-tex">{\displaystyle k[x,y]}</annotation>

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</math><img src="./e8462ca2b7942a5b5d38beed97662788b27b568a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.024ex; height:2.843ex;" alt="{\displaystyle k[x,y]}" loading="lazy"> consider the <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by the <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}-x^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle y^{2}-x^{3}}</annotation>

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</math><img src="./d7ed786b806e6cc994a6039e8342fbcea3e16622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.439ex; height:3.009ex;" alt="{\displaystyle y^{2}-x^{3}}" loading="lazy"> and form the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> of the <a href="Quotient_ring" title="Quotient ring">quotient ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x,y]/(y^{2}-x^{3})}">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle k[x,y]/(y^{2}-x^{3})}</annotation>

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</math><img src="./796d355c09d7f883ced5786aee36d0abe41500dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.435ex; height:3.176ex;" alt="{\displaystyle k[x,y]/(y^{2}-x^{3})}" loading="lazy">. This is a function field of one variable over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">; it can also be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(x)({\sqrt {x^{3}}})}">

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<annotation encoding="application/x-tex">{\displaystyle k(x)({\sqrt {x^{3}}})}</annotation>

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</math><img src="./10882c37ad03bef4da1c243da2e41ec9fabbe59a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.867ex; height:3.509ex;" alt="{\displaystyle k(x)({\sqrt {x^{3}}})}" loading="lazy"> (with degree 2 over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(x)}">

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

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</math><img src="./6e58c3a9efef5b14a9ba8f2bc27baf8bac2c3f25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.35ex; height:2.843ex;" alt="{\displaystyle k(x)}" loading="lazy">) or as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(y)({\sqrt[{3}]{y^{2}}})}">

<mroot>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle k(y)({\sqrt[{3}]{y^{2}}})}</annotation>

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</math><img src="./54318206c4286a824141d8c82c70b5542949e8f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:10.524ex; height:4.843ex;" alt="{\displaystyle k(y)({\sqrt[{3}]{y^{2}}})}" loading="lazy"> (with degree 3 over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(y)}">

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<annotation encoding="application/x-tex">{\displaystyle k(y)}</annotation>

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</math><img src="./7bd5e581f80865616939ad1f42059e7ed2078e8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle k(y)}" loading="lazy">). We see that the degree of an algebraic function field is not a well-defined notion.

<div class="mw-heading mw-heading2"><h2 id="Category_structure">Category structure</h2></div>

The algebraic function fields over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> form a <a href="Category_(mathematics)" title="Category (mathematics)">category</a>; the <a href="Morphism_(category_theory)" class="mw-redirect" title="Morphism (category theory)">morphisms</a> from function field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> are the <a href="Ring_homomorphism" title="Ring homomorphism">ring homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:K\to L}">

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<annotation encoding="application/x-tex">{\displaystyle f:K\to L}</annotation>

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</math><img src="./984f1e4e3c32ef49e18ee29d77533fd2489defae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.479ex; height:2.509ex;" alt="{\displaystyle f:K\to L}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)=a}">

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

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</math><img src="./70ea63e918d829b0f0b20efc54b2b6d739af5143.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.646ex; height:2.843ex;" alt="{\displaystyle f(a)=a}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. All these morphisms are <a href="Injective_function" title="Injective function">injective</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is a function field over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> variables, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is a function field in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> variables, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>m}">

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

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</math><img src="./e64e2a4a5b6cb58f1553c6a65551b4898bb82403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle n>m}" loading="lazy">, then there are no morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Function_fields_arising_from_varieties,_curves_and_Riemann_surfaces">Function fields arising from varieties, curves and Riemann surfaces</h2></div>

The <a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field of an algebraic variety</a> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is an algebraic function field of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> variables over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</mrow>

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.

Two varieties are <a href="Birational_geometry" title="Birational geometry">birationally equivalent</a> if and only if their function fields are isomorphic (but note that non-<a href="Morphism_of_varieties" class="mw-redirect" title="Morphism of varieties">isomorphic</a> varieties may have the same function field). Assigning to each variety its function field yields a <a href="Equivalence_of_categories" title="Equivalence of categories">duality</a> (contravariant equivalence) between the category of varieties over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> (with <a href="Rational_mapping" title="Rational mapping">dominant rational maps</a> as morphisms) and the category of algebraic function fields over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. The varieties considered here are to be taken in the <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> sense; they need not have any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-rational points, like the curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+1=0}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}+1=0}</annotation>

</semantics>

</math><img src="./9db8c3c2c1f343e444aa4d9286d24242cdb2319d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.703ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+1=0}" loading="lazy"> defined over the real numbers.

The case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"> (irreducible algebraic curves in the <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> sense) is especially important, since every function field of one variable over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> arises as the function field of a uniquely defined <a href="Regular_scheme" title="Regular scheme">regular</a> (i.e. non-singular) projective irreducible algebraic curve over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves (with <a href="Glossary_of_scheme_theory" class="mw-redirect" title="Glossary of scheme theory">dominant</a> <a href="Regular_map_(algebraic_geometry)" class="mw-redirect" title="Regular map (algebraic geometry)">regular maps</a> as morphisms) and the category of function fields of one variable over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.

The field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(X)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b1f56579142abfdec868307b665356a273f2b251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.232ex; height:2.843ex;" alt="{\displaystyle M(X)}" loading="lazy"> of <a href="Meromorphic_function" title="Meromorphic function">meromorphic functions</a> defined on a connected <a href="Riemann_surface" title="Riemann surface">Riemann surface</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a function field of one variable over the <a href="Complex_number" title="Complex number">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">. In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(X)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b1f56579142abfdec868307b665356a273f2b251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.232ex; height:2.843ex;" alt="{\displaystyle M(X)}" loading="lazy"> yields a duality between the category of compact connected Riemann surfaces (with non-constant <a href="Holomorphic" class="mw-redirect" title="Holomorphic">holomorphic</a> maps as morphisms) and function fields of one variable over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">. A similar correspondence exists between compact connected <a href="Klein_surface" title="Klein surface">Klein surfaces</a> and function fields in one variable over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Number_fields_and_finite_fields">Number fields and finite fields</h2></div>

The <a href="Function_field_analogy" class="mw-redirect" title="Function field analogy">function field analogy</a> states that almost all theorems on <a href="Number_field" class="mw-redirect" title="Number field">number fields</a> have a counterpart on function fields of one variable over a <a href="Finite_field" title="Finite field">finite field</a>, and these counterparts are frequently easier to prove (see <a href="Prime_number_theorem#Analogue_for_irreducible_polynomials_over_a_finite_field" title="Prime number theorem">analogue for irreducible polynomials over a finite field</a>). In the context of this analogy, both number fields and function fields over finite fields are usually called "<a href="Global_field" title="Global field">global fields</a>".

The study of function fields over a finite field has applications in <a href="Cryptography" title="Cryptography">cryptography</a> and <a href="Error_correcting_code" class="mw-redirect" title="Error correcting code">error correcting codes</a>. For example, the function field of an <a href="Elliptic_curve" title="Elliptic curve">elliptic curve</a> over a finite field (an important mathematical tool for <a href="Public_key_cryptography" class="mw-redirect" title="Public key cryptography">public key cryptography</a>) is an algebraic function field.

Function fields over the field of <a href="Rational_number" title="Rational number">rational numbers</a> play also an important role in solving <a href="Inverse_Galois_problem" title="Inverse Galois problem">inverse Galois problems</a>.

<div class="mw-heading mw-heading2"><h2 id="Field_of_constants">Field of constants</h2></div>

Given any algebraic function field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, we can consider the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> which are <a href="Algebraic_element" title="Algebraic element">algebraic</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. These elements form a field, known as the field of constants of the algebraic function field.

For instance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} (x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} (x)}</annotation>

</semantics>

</math><img src="./db8e92fab0e6edb58cba00edaa5f69d287f46898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} (x)}" loading="lazy"> is a function field of one variable over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">; its field of constants is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Valuations_and_places">Valuations and places</h2></div>

Key tools to study algebraic function fields are <a href="Absolute_value_(algebra)" title="Absolute value (algebra)">absolute values, valuations, places</a> and their completions.

Given an algebraic function field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/k}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dea0107b51200e74c3cd319d023acdb194ebd506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.44ex; height:2.843ex;" alt="{\displaystyle K/k}" loading="lazy"> of one variable, we define the notion of a valuation ring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/k}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dea0107b51200e74c3cd319d023acdb194ebd506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.44ex; height:2.843ex;" alt="{\displaystyle K/k}" loading="lazy">: this is a <a href="Subring" title="Subring">subring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d6ae2ed4058fb748a183d9ada8aea50a00d0c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> that contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> and is different from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">, and such that for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\mathcal {O}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8bb49d6dbd7013005ba5c0c9563eb99bbc21f7f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.02ex; height:2.176ex;" alt="{\displaystyle x\in {\mathcal {O}}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}\in {\mathcal {O}}}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{-1}\in {\mathcal {O}}}</annotation>

</semantics>

</math><img src="./73831d7e91aa8a09a8c6d7e9de3bac4761d503bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.353ex; height:2.676ex;" alt="{\displaystyle x^{-1}\in {\mathcal {O}}}" loading="lazy">. Each such valuation ring is a <a href="Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation ring</a> and its maximal ideal is called a place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/k}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dea0107b51200e74c3cd319d023acdb194ebd506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.44ex; height:2.843ex;" alt="{\displaystyle K/k}" loading="lazy">.

A discrete valuation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/k}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dea0107b51200e74c3cd319d023acdb194ebd506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.44ex; height:2.843ex;" alt="{\displaystyle K/k}" loading="lazy"> is a <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v:K\to \mathbb {Z} \cup \{\infty \}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v:K\to \mathbb {Z} \cup \{\infty \}}</annotation>

</semantics>

</math><img src="./9045f3794a893d284b78d5fbc71b44f819b23a6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.526ex; height:2.843ex;" alt="{\displaystyle v:K\to \mathbb {Z} \cup \{\infty \}}" loading="lazy"> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./93db9c4784bc82a2dfe9ea58ea1167080010e574.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.426ex; height:2.509ex;" alt="{\displaystyle x,y\in K}" loading="lazy">,

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(xy)=v(x)+v(y)}</annotation>

</semantics>

</math><img src="./3e3afc1ef604537b84d68e9bda4435ca54aab9bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.72ex; height:2.843ex;" alt="{\displaystyle v(xy)=v(x)+v(y)}" loading="lazy">,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(x+y)\geq \min\{v(x),v(y)\}}</annotation>

</semantics>

</math><img src="./28b6e72b75388037989d7b76c41ccb566e621efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.954ex; height:2.843ex;" alt="{\displaystyle v(x+y)\geq \min\{v(x),v(y)\}}" loading="lazy">,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(x)=\infty \iff x=0}</annotation>

</semantics>

</math><img src="./634fa4259c701ac713cad0d4c928332a101bc0d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.177ex; height:2.843ex;" alt="{\displaystyle v(x)=\infty \iff x=0}" loading="lazy"></li></ul>

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(a)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(a)=0}</annotation>

</semantics>

</math><img src="./cc952e1a8e8d54a03ef4236483525b813fecf9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.428ex; height:2.843ex;" alt="{\displaystyle v(a)=0}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in k\setminus \{0\}}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a\in k\setminus \{0\}}</annotation>

</semantics>

</math><img src="./0e3a0b80f4c12fd803b6b6f0eaa489cda537a6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.964ex; height:2.843ex;" alt="{\displaystyle a\in k\setminus \{0\}}" loading="lazy">.

There are natural bijective correspondences between the set of valuation rings of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/k}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dea0107b51200e74c3cd319d023acdb194ebd506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.44ex; height:2.843ex;" alt="{\displaystyle K/k}" loading="lazy">, the set of places of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/k}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dea0107b51200e74c3cd319d023acdb194ebd506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.44ex; height:2.843ex;" alt="{\displaystyle K/k}" loading="lazy">, and the set of discrete valuations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/k}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dea0107b51200e74c3cd319d023acdb194ebd506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.44ex; height:2.843ex;" alt="{\displaystyle K/k}" loading="lazy">. These sets can be given a natural <a href="Topology" title="Topology">topological</a> structure: the <a href="Zariski%E2%80%93Riemann_space" title="Zariski–Riemann space">Zariski–Riemann space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/k}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dea0107b51200e74c3cd319d023acdb194ebd506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.44ex; height:2.843ex;" alt="{\displaystyle K/k}" loading="lazy">.

<ul><li><a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field of an algebraic variety</a></li>

<li><a href="Function_field_(scheme_theory)" title="Function field (scheme theory)">function field (scheme theory)</a></li>

<li><a href="Algebraic_function" title="Algebraic function">algebraic function</a></li>

<li><a href="Drinfeld_module" title="Drinfeld module">Drinfeld module</a></li></ul>

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

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</style><cite id="CITEREFGabriel_DanielVilla_Salvador2007" class="citation book cs1">Gabriel Daniel & Villa Salvador (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RmKpEUltmQIC">Topics in the Theory of Algebraic Function Fields</a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780817645151</bdi>.</cite>

</li>

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