Field trace

Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can be viewed as a vector space over K. Multiplication by α, an element of L,

${\displaystyle m_{\alpha }:L\to L{\text{ given by }}m_{\alpha }(x)=\alpha x}$,

is a K-linear transformation of this vector space into itself. The trace, Tr_L/K(α), is defined as the (linear algebra) trace of this linear transformation.[1]

For α in L, let σ₁(α), ..., σ_n(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K), then

${\displaystyle \operatorname {Tr} _{L/K}(\alpha )=[L:K(\alpha )]\sum _{j=1}^{n}\sigma _{j}(\alpha )}$.

If L/K is separable then each root appears only once[2] (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 Galois extension and α is in L, then the trace of α is the sum of all the Galois conjugates of α,[1] i.e.,

${\displaystyle \operatorname {Tr} _{L/K}(\alpha )=\sum _{\sigma \in \operatorname {Gal} (L/K)}\sigma (\alpha ),}$

where Gal(L/K) denotes the Galois group of L/K.

Let ${\displaystyle L=\mathbb {Q} ({\sqrt {d}})}$ be a quadratic extension of ${\displaystyle \mathbb {Q} }$. Then a basis of ${\displaystyle L/\mathbb {Q} {\text{ is }}\{1,{\sqrt {d}}\}.}$ If ${\displaystyle \alpha =a+b{\sqrt {d}}}$ then the matrix of ${\displaystyle m_{\alpha }}$ is:

${\displaystyle \left[{\begin{matrix}a&bd\\b&a\end{matrix}}\right]}$,

and so, ${\displaystyle \operatorname {Tr} _{L/\mathbb {Q} }(\alpha )=[L:\mathbb {Q} (\alpha )]\left(\sigma _{1}(\alpha )+\sigma _{2}(\alpha )\right)=1\times \left(\sigma _{1}(\alpha )+{\overline {\sigma _{1}}}(\alpha )\right)=a+b{\sqrt {d}}+a-b{\sqrt {d}}=2a}$.[1] The minimal polynomial of α is X² − 2a X + a² − d b².

Several properties of the trace function hold for any finite extension.[3]

The trace Tr_L/K : L → K is a K-linear map (a K-linear functional), that is

${\displaystyle \operatorname {Tr} _{L/K}(\alpha a+\beta b)=\alpha \operatorname {Tr} _{L/K}(a)+\beta \operatorname {Tr} _{L/K}(b){\text{ for all }}\alpha ,\beta \in K}$.

If α ∈ K then ${\displaystyle \operatorname {Tr} _{L/K}(\alpha )=[L:K]\alpha .}$

Additionally, trace behaves well in towers of fields: if M is a finite extension of L, then the trace from M to K is just the composition of the trace from M to L with the trace from L to K, i.e.

${\displaystyle \operatorname {Tr} _{M/K}=\operatorname {Tr} _{L/K}\circ \operatorname {Tr} _{M/L}}$.

Let L = GF(qⁿ) be a finite extension of a finite field K = GF(q). Since L/K is a Galois extension, if α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.[4]

${\displaystyle \operatorname {Tr} _{L/K}(\alpha )=\alpha +\alpha ^{q}+\cdots +\alpha ^{q^{n-1}}}$.

In this setting we have the additional properties:[5]

• ${\displaystyle \operatorname {Tr} _{L/K}(a^{q})=\operatorname {Tr} _{L/K}(a){\text{ for }}a\in L}$.
• For any ${\displaystyle \alpha \in K}$, there are exactly ${\displaystyle q^{n-1}}$ elements ${\displaystyle b\in L}$ with ${\displaystyle \operatorname {Tr} _{L/K}(b)=\alpha }$.

Theorem.[6] For b ∈ L, let F_b be the map ${\displaystyle a\mapsto \operatorname {Tr} _{L/K}(ba).}$ Then F_b ≠ F_c if b ≠ c. Moreover, the K-linear transformations from L to K are exactly the maps of the form F_b as b varies over the field L.

When K is the prime subfield of L, the trace is called the absolute trace and otherwise it is a relative trace.[4]

When L/K is separable, the trace provides a duality theory via the trace form: the map from L × L to K sending (x, y) to Tr_L/K(xy) is a nondegenerate, symmetric, bilinear form 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 algebraic number theory in the theory of the different ideal.

The trace form for a finite degree field extension L/K has non-negative signature for any field ordering of K.[8] The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.[8]

If L/K is an inseparable extension, then the trace form is identically 0.[9]

• Field norm
• Reduced trace

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• Isaacs, I.M. (1994), Algebra, A Graduate Course, Brooks/Cole Publishing
• Lidl, Rudolf; Niederreiter, Harald (1997) [1983], Finite Fields, Encyclopedia of Mathematics and its Applications, 20 (Second ed.), Cambridge University Press, ISBN 0-521-39231-4, Zbl 0866.11069
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6
• Roman, Steven (2006), Field theory, Graduate Texts in Mathematics, 158 (Second ed.), Springer, Chapter 8, ISBN 978-0-387-27677-9, Zbl 1172.12001
• Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 978-0-13-087868-7

• Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017.
• Section VI.5 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001