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Field trace

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Trace_(disambiguation)" class="mw-redirect mw-disambig" title="Trace (disambiguation)">Trace (disambiguation)</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, the field trace is a particular <a href="Function_(mathematics)" title="Function (mathematics)">function</a> defined with respect to a <a href="Finite_extension" class="mw-redirect" title="Finite extension">finite</a> <a href="Field_extension" title="Field extension">field extension</a> L/K, which is a <a href="Linear_map" title="Linear map">K-linear map</a> from L onto K.

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

Let K be a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> and L a finite extension (and hence an <a href="Algebraic_extension" title="Algebraic extension">algebraic extension</a>) of K. L can be viewed as a <a href="Vector_space" title="Vector space">vector space</a> over K. Multiplication by α, an element of L,

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<annotation encoding="application/x-tex">{\displaystyle m_{\alpha }:L\to L{\text{ given by }}m_{\alpha }(x)=\alpha x}</annotation>

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</math><img src="./60135b6d873d8e8dba9916f1139b3db70a03a1c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.044ex; height:2.843ex;" alt="{\displaystyle m_{\alpha }:L\to L{\text{ given by }}m_{\alpha }(x)=\alpha x}" loading="lazy">,</dd></dl>

is a K-<a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> of this vector space into itself. The trace, TrL/K(α), is defined as the <a href="Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> (in the <a href="Linear_algebra" title="Linear algebra">linear algebra</a> sense) of this linear transformation.<a href="#cite_note-ROT940-1">[1]</a>

For α in L, let σ1(α), ..., σn(α) be the <a href="Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">roots</a> (counted with multiplicity) of the <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> of α over K (in some extension field of K). Then

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{L/K}(\alpha )=[L:K(\alpha )]\sum _{j=1}^{n}\sigma _{j}(\alpha ).}</annotation>

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</math><img src="./aa4f423368097975441959e7544cd874456e3bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.106ex; height:7.176ex;" alt="{\displaystyle \operatorname {Tr} _{L/K}(\alpha )=[L:K(\alpha )]\sum _{j=1}^{n}\sigma _{j}(\alpha ).}" loading="lazy"></dd></dl>

If L/K is <a href="Separable_extension" title="Separable extension">separable</a> then each root appears only once<a href="#cite_note-2">[2]</a> (however this does not mean the coefficient above is one; for example if α is the identity element 1 of K then the trace is [L:K] times 1).

More particularly, if L/K is a <a href="Galois_extension" title="Galois extension">Galois extension</a> and α is in L, then the trace of α is the sum of all the <a href="Galois_conjugate" class="mw-redirect" title="Galois conjugate">Galois conjugates</a> of α,<a href="#cite_note-ROT940-1">[1]</a> i.e.,

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{L/K}(\alpha )=\sum _{\sigma \in \operatorname {Gal} (L/K)}\sigma (\alpha ),}</annotation>

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</math><img src="./23ba10446d735aa06317e2e6f6a004549128ade6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.567ex; height:6.009ex;" alt="{\displaystyle \operatorname {Tr} _{L/K}(\alpha )=\sum _{\sigma \in \operatorname {Gal} (L/K)}\sigma (\alpha ),}" loading="lazy"></dd></dl>

where Gal(L/K) denotes the <a href="Galois_group" title="Galois group">Galois group</a> of L/K.

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

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\mathbb {Q} ({\sqrt {d}})}">

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<annotation encoding="application/x-tex">{\displaystyle L=\mathbb {Q} ({\sqrt {d}})}</annotation>

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</math><img src="./6dfb02b1c0514b0dea06b56c4aea1ca23d02ef00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.45ex; height:3.176ex;" alt="{\displaystyle L=\mathbb {Q} ({\sqrt {d}})}" loading="lazy"> be a <a href="Quadratic_extension" class="mw-redirect" title="Quadratic extension">quadratic extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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

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</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">. Then a <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/\mathbb {Q} }">

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

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</math><img src="./3a1ebcca5c1b26fbd2e060ad6cc8e820e01baca0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.553ex; height:2.843ex;" alt="{\displaystyle L/\mathbb {Q} }" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,{\sqrt {d}}\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,{\sqrt {d}}\}.}</annotation>

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</math><img src="./e60f6fcbb3aab32f37b517f1fab0d14341ded56e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.32ex; height:3.176ex;" alt="{\displaystyle \{1,{\sqrt {d}}\}.}" loading="lazy"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =a+b{\sqrt {d}}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =a+b{\sqrt {d}}}</annotation>

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</math><img src="./95a06f1298047b0936fda5a54c3ff1bb16fadf34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.806ex; height:3.009ex;" alt="{\displaystyle \alpha =a+b{\sqrt {d}}}" loading="lazy"> then the <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\alpha }}">

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

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</math><img src="./46cbcede3b6470d7e97fb926da5a426ac307f16d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.325ex; height:2.009ex;" alt="{\displaystyle m_{\alpha }}" loading="lazy"> is:

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<annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}a&bd\\b&a\end{matrix}}\right]}</annotation>

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</math><img src="./9874d45b8c9da9e90b57f725cad7f3adef701496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.972ex; height:6.176ex;" alt="{\displaystyle \left[{\begin{matrix}a&bd\\b&a\end{matrix}}\right]}" loading="lazy">,</dd></dl>

and so, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tr} _{L/\mathbb {Q} }(\alpha )=[L:\mathbb {Q} (\alpha )]\left(\sigma _{1}(\alpha )+\sigma _{2}(\alpha )\right)=1\times \left(\sigma _{1}(\alpha )+{\overline {\sigma _{1}}}(\alpha )\right)=a+b{\sqrt {d}}+a-b{\sqrt {d}}=2a}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{L/\mathbb {Q} }(\alpha )=[L:\mathbb {Q} (\alpha )]\left(\sigma _{1}(\alpha )+\sigma _{2}(\alpha )\right)=1\times \left(\sigma _{1}(\alpha )+{\overline {\sigma _{1}}}(\alpha )\right)=a+b{\sqrt {d}}+a-b{\sqrt {d}}=2a}</annotation>

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</math><img src="./50c997744da837647ba9d193cf5b6d75fc5d4526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:89.841ex; height:3.509ex;" alt="{\displaystyle \operatorname {Tr} _{L/\mathbb {Q} }(\alpha )=[L:\mathbb {Q} (\alpha )]\left(\sigma _{1}(\alpha )+\sigma _{2}(\alpha )\right)=1\times \left(\sigma _{1}(\alpha )+{\overline {\sigma _{1}}}(\alpha )\right)=a+b{\sqrt {d}}+a-b{\sqrt {d}}=2a}" loading="lazy">.<a href="#cite_note-ROT940-1">[1]</a> The minimal polynomial of α is X2 − 2a X + (a2 − db2).

<div class="mw-heading mw-heading2"><h2 id="Properties_of_the_trace">Properties of the trace</h2></div>

Several properties of the trace function hold for any finite extension.<a href="#cite_note-3">[3]</a>

The trace TrL/K : L → K is a K-<a href="Linear_map" title="Linear map">linear map</a> (a K-linear functional), that is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{L/K}(\alpha a+\beta b)=\alpha \operatorname {Tr} _{L/K}(a)+\beta \operatorname {Tr} _{L/K}(b){\text{ for all }}\alpha ,\beta \in K}</annotation>

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</math><img src="./a244e5b24e6a9e3783b26738b073d475b336bc75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:59.492ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tr} _{L/K}(\alpha a+\beta b)=\alpha \operatorname {Tr} _{L/K}(a)+\beta \operatorname {Tr} _{L/K}(b){\text{ for all }}\alpha ,\beta \in K}" loading="lazy">.</dd></dl>

If α ∈ K then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tr} _{L/K}(\alpha )=[L:K]\alpha .}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{L/K}(\alpha )=[L:K]\alpha .}</annotation>

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</math><img src="./5df17fe0dbc3a0b9a696db1a90d2e8aa42b7ff84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.633ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tr} _{L/K}(\alpha )=[L:K]\alpha .}" loading="lazy">

Additionally, trace behaves well in <a href="Tower_of_fields" title="Tower of fields">towers of fields</a>: if M is a finite extension of L, then the trace from M to K is just the <a href="Function_composition" title="Function composition">composition</a> of the trace from M to L with the trace from L to K, i.e.

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{M/K}=\operatorname {Tr} _{L/K}\circ \operatorname {Tr} _{M/L}}</annotation>

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</math><img src="./20faacc5cb513b024b5cb68216d2a3e9a941756b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.581ex; height:3.009ex;" alt="{\displaystyle \operatorname {Tr} _{M/K}=\operatorname {Tr} _{L/K}\circ \operatorname {Tr} _{M/L}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Finite_fields">Finite fields</h2></div>

Let L = GF(qn) be a finite extension of a <a href="Finite_field" title="Finite field">finite field</a> K = GF(q). Since L/K is a <a href="Galois_extension" title="Galois extension">Galois extension</a>, if α is in L, then the trace of α is the sum of all the <a href="Galois_conjugate" class="mw-redirect" title="Galois conjugate">Galois conjugates</a> of α, i.e.<a href="#cite_note-LN54-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{L/K}(\alpha )=\alpha +\alpha ^{q}+\cdots +\alpha ^{q^{n-1}}.}</annotation>

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</math><img src="./38bd3936a92707c4a72f6916d0e37fd6828dcf90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.628ex; height:3.843ex;" alt="{\displaystyle \operatorname {Tr} _{L/K}(\alpha )=\alpha +\alpha ^{q}+\cdots +\alpha ^{q^{n-1}}.}" loading="lazy"></dd></dl>

In this setting we have the additional properties:<a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{L/K}(a^{q})=\operatorname {Tr} _{L/K}(a){\text{ for }}a\in L}</annotation>

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</math><img src="./e52450b531a326a5fb9ffc688f687dd1b86f9ab5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.214ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tr} _{L/K}(a^{q})=\operatorname {Tr} _{L/K}(a){\text{ for }}a\in L}" loading="lazy">.</li>

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

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</math><img src="./042c3f6ca6ee2a5c044168ae099e772e34313cff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.398ex; height:3.009ex;" alt="{\displaystyle q^{n-1}}" loading="lazy"> elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in L}">

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

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</math><img src="./23f8717d0feda0c7326628ea9e1a59f4b40e47a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.421ex; height:2.176ex;" alt="{\displaystyle b\in L}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tr} _{L/K}(b)=\alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{L/K}(b)=\alpha }</annotation>

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Theorem.<a href="#cite_note-LN56-6">[6]</a> For b ∈ L, let Fb be the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mapsto \operatorname {Tr} _{L/K}(ba).}">

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<annotation encoding="application/x-tex">{\displaystyle a\mapsto \operatorname {Tr} _{L/K}(ba).}</annotation>

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</math><img src="./eb0af5f534ecf5b8e4445904d37cb856acd9b936.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.751ex; height:3.176ex;" alt="{\displaystyle a\mapsto \operatorname {Tr} _{L/K}(ba).}" loading="lazy"> Then Fb ≠ Fc if b ≠ c. Moreover, the K-linear transformations from L to K are exactly the maps of the form Fb as b varies over the field L.

When K is the <a href="Prime_subfield" class="mw-redirect" title="Prime subfield">prime subfield</a> of L, the trace is called the absolute trace and otherwise it is a relative trace.<a href="#cite_note-LN54-4">[4]</a>

<div class="mw-heading mw-heading3"><h3 id="Application">Application</h3></div>

A <a href="Quadratic_equation" title="Quadratic equation">quadratic equation</a>, ax2 + bx + c = 0 with a ≠ 0, and coefficients in the finite field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GF} (q)=\mathbb {F} _{q}}">

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

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Consider the quadratic equation ax2 + bx + c = 0 with coefficients in the finite field GF(2h).<a href="#cite_note-7">[7]</a> If b = 0 then this equation has the unique solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\sqrt {\frac {c}{a}}}}">

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

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<annotation encoding="application/x-tex">{\displaystyle y^{2}+y+\delta =0,{\text{ where }}\delta ={\frac {ac}{b^{2}}}.}</annotation>

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</math><img src="./04f6b3e2f7c6417b348f5ae7a75cb3f021720738.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.37ex; height:5.009ex;" alt="{\displaystyle y^{2}+y+\delta =0,{\text{ where }}\delta ={\frac {ac}{b^{2}}}.}" loading="lazy"></dd></dl>

This equation has two solutions in GF(q) <a href="If_and_only_if" title="If and only if">if and only if</a> the absolute trace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tr} _{GF(q)/GF(2)}(\delta )=0.}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{GF(q)/GF(2)}(\delta )=0.}</annotation>

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</math><img src="./a3b6765114fd288032101fbed7305779f2ae820c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.592ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tr} _{GF(q)/GF(2)}(\delta )=0.}" loading="lazy"> In this case, if y = s is one of the solutions, then y = s + 1 is the other. Let k be any element of GF(q) with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tr} _{GF(q)/GF(2)}(k)=1.}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{GF(q)/GF(2)}(k)=1.}</annotation>

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</math><img src="./63b3dbff3a03c8187051ab779810a0f416909fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.754ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tr} _{GF(q)/GF(2)}(k)=1.}" loading="lazy"> Then a solution to the equation is given by:

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<annotation encoding="application/x-tex">{\displaystyle y=s=k\delta ^{2}+(k+k^{2})\delta ^{4}+\ldots +(k+k^{2}+\ldots +k^{2^{h-2}})\delta ^{2^{h-1}}.}</annotation>

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</math><img src="./a4864351c099fc6013349631b18efd97515953cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.068ex; height:3.509ex;" alt="{\displaystyle y=s=k\delta ^{2}+(k+k^{2})\delta ^{4}+\ldots +(k+k^{2}+\ldots +k^{2^{h-2}})\delta ^{2^{h-1}}.}" loading="lazy"></dd></dl>

When h = 2m' + 1, a solution is given by the simpler expression:

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<annotation encoding="application/x-tex">{\displaystyle y=s=\delta +\delta ^{2^{2}}+\delta ^{2^{4}}+\ldots +\delta ^{2^{2m}}.}</annotation>

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</math><img src="./f25e4ac66bc8cd2cf84027be2db0463c2e311463.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.213ex; height:3.343ex;" alt="{\displaystyle y=s=\delta +\delta ^{2^{2}}+\delta ^{2^{4}}+\ldots +\delta ^{2^{2m}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Trace_form">Trace form</h2></div>

When L/K is separable, the trace provides a <a href="Duality_theory" class="mw-redirect" title="Duality theory">duality theory</a> via the trace form: the map from L × L to K sending (x, y) to TrL/K(xy) is a <a href="Nondegenerate_form" class="mw-redirect" title="Nondegenerate form">nondegenerate</a>, <a href="Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric bilinear form</a> called the trace form. If L/K is a Galois extension, the trace form is invariant with respect to the Galois group.

The trace form is used in <a href="Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a> in the theory of the <a href="Different_ideal" title="Different ideal">different ideal</a>.

The trace form for a finite degree field extension L/K has non-negative <a href="Signature_(quadratic_form)" class="mw-redirect" title="Signature (quadratic form)">signature</a> for any <a href="Field_ordering" class="mw-redirect" title="Field ordering">field ordering</a> of K.<a href="#cite_note-L38-8">[8]</a> The <a href="Converse_(logic)" title="Converse (logic)">converse</a>, that every <a href="Witt_ring_(forms)" class="mw-redirect" title="Witt ring (forms)">Witt equivalence</a> class with non-negative signature contains a trace form, is true for <a href="Algebraic_number_field" title="Algebraic number field">algebraic number fields</a> K.<a href="#cite_note-L38-8">[8]</a>

If L/K is an <a href="Inseparable_extension" class="mw-redirect" title="Inseparable extension">inseparable extension</a>, then the trace form is identically 0.<a href="#cite_note-9">[9]</a>

<ul><li><a href="Field_norm" title="Field norm">Field norm</a></li>

<li><a href="Reduced_trace" class="mw-redirect" title="Reduced trace">Reduced trace</a></li></ul>

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

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<li id="cite_note-ROT940-1">^ <a href="#cite_ref-ROT940_1-0">a</a> <a href="#cite_ref-ROT940_1-1">b</a> <a href="#cite_ref-ROT940_1-2">c</a> <a href="#CITEREFRotman2002">Rotman 2002</a>, p. 940

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFRotman2002">Rotman 2002</a>, p. 941

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFRoman2006">Roman 2006</a>, p. 151

</li>

<li id="cite_note-LN54-4">^ <a href="#cite_ref-LN54_4-0">a</a> <a href="#cite_ref-LN54_4-1">b</a> <a href="#CITEREFLidlNiederreiter1997">Lidl & Niederreiter 1997</a>, p.54

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFMullenPanario2013">Mullen & Panario 2013</a>, p. 21

</li>

<li id="cite_note-LN56-6"><a href="#cite_ref-LN56_6-0">^</a> <a href="#CITEREFLidlNiederreiter1997">Lidl & Niederreiter 1997</a>, p.56

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFHirschfeld1979">Hirschfeld 1979</a>, pp. 3-4

</li>

<li id="cite_note-L38-8">^ <a href="#cite_ref-L38_8-0">a</a> <a href="#cite_ref-L38_8-1">b</a> Lorenz (2008) p.38

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFIsaacs1994">Isaacs 1994</a>, p. 369 as footnoted in <a href="#CITEREFRotman2002">Rotman 2002</a>, p. 943

</li>

</ol></div>

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

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</style><cite id="CITEREFHirschfeld1979" class="citation cs2">Hirschfeld, J.W.P. (1979), <a rel="nofollow" class="external text" href="https://archive.org/details/projectivegeomet0000hirs">Projective Geometries over Finite Fields</a>, Oxford Mathematical Monographs, Oxford University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-853526-0</bdi></cite></li>

<li><cite id="CITEREFIsaacs1994" class="citation cs2">Isaacs, I.M. (1994), Algebra, A Graduate Course, Brooks/Cole Publishing</cite></li>

<li><cite id="CITEREFLidlNiederreiter1997" class="citation cs2">Lidl, Rudolf; <a href="Harald_Niederreiter" title="Harald Niederreiter">Niederreiter, Harald</a> (1997) [1983], <a rel="nofollow" class="external text" href="https://archive.org/details/finitefields0000lidl_a8r3">Finite Fields</a>, Encyclopedia of Mathematics and its Applications, vol. 20 (Second ed.), <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-39231-4</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:0866.11069">0866.11069</a></cite></li>

<li><cite id="CITEREFLorenz2008" class="citation book cs1">Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-72487-4</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:1130.12001">1130.12001</a>.</cite></li>

<li><cite id="CITEREFMullenPanario2013" class="citation cs2">Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4398-7378-6</bdi></cite></li>

<li><cite id="CITEREFRoman2006" class="citation cs2">Roman, Steven (2006), Field theory, Graduate Texts in Mathematics, vol. 158 (Second ed.), Springer, Chapter 8, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-27677-9</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:1172.12001">1172.12001</a></cite></li>

<li><cite id="CITEREFRotman2002" class="citation cs2">Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-087868-7</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFConnerPerlis1984" class="citation book cs1">Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. Vol. 2. World Scientific. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9971-966-05-0</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:0551.10017">0551.10017</a>.</cite></li>

<li>Section VI.5 of <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>, <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:0984.00001">0984.00001</a></cite></li></ul></div><div><div class="zim-footer">

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