Trace (field theory)

Trace ( eng. Trace ) - displaying the elements of a finite field extension $E\supset K$ ${\ displaystyle E \ supset K}$ $E \ supset K$ in the original field K , defined as follows:

Let E be a finite extension of degree K $n=[E:K]$ ${\ displaystyle n = [E: K]}$ $n = [E: K]$ , $\alpha \in E$ ${\ displaystyle \ alpha \ in E}$ $\ alpha \ in E$ Is an element of the field E. Since E is a vector space over the field K , this element defines a linear transformation $x\mapsto \alpha x$ ${\ displaystyle x \ mapsto \ alpha x}$ $x \ mapsto \ alpha x$ . A matrix can be associated with this transformation in some basis. The trace of this matrix is called the trace of the element α . Since in a different basis a similar matrix with the same trace will correspond to this mapping, the trace does not depend on the choice of basis, that is, each trace element is uniquely associated with its trace. It is designated ${\text{Tr}}_{K}^{E}(\alpha )$ ${\ displaystyle {\ text {Tr}} _ {K} ^ {E} (\ alpha)}$ ${\ text {Tr}} _ {K} ^ {E} (\ alpha)$ or, if it’s clear what kind of extension it’s ${\text{Tr}}(\alpha )$ ${\ displaystyle {\ text {Tr}} (\ alpha)}$ ${\ text {Tr}} (\ alpha)$ .

Content

1 Trace Properties
2 Expression of a trace in terms of automorphisms E over K
3 Example
4 See also
5 Literature

Trace Properties

${\text{Tr}}(\alpha +\beta )={\text{Tr}}(\alpha )+{\text{Tr}}(\beta )$ ${\ displaystyle {\ text {Tr}} (\ alpha + \ beta) = {\ text {Tr}} (\ alpha) + {\ text {Tr}} (\ beta)}$
${\text{Tr}}(c\alpha )=c{\text{Tr}}(\alpha )$ ${\ displaystyle {\ text {Tr}} (c \ alpha) = c {\ text {Tr}} (\ alpha)}$ at $c\in K$ ${\ displaystyle c \ in K}$
If E is a separable extension , then ${\text{Tr}}_{K}^{E}$ ${\ displaystyle {\ text {Tr}} _ {K} ^ {E}}$ - nonzero functional, if inseparable, then ${\text{Tr}}_{K}^{E}=0$ ${\ displaystyle {\ text {Tr}} _ {K} ^ {E} = 0}$ .
The trace is transitive, i.e. for the extension chain $K\subset E\subset F$ ${\ displaystyle K \ subset E \ subset F}$ we have ${\text{Tr}}_{K}^{E}({\text{Tr}}_{E}^{F}(\alpha ))={\text{Tr}}_{K}^{F}(\alpha )$ ${\ displaystyle {\ text {Tr}} _ {K} ^ {E} ({\ text {Tr}} _ {E} ^ {F} (\ alpha)) = {\ text {Tr}} _ {K } ^ {F} (\ alpha)}$
If $E=K(\alpha )$ ${\ displaystyle E = K (\ alpha)}$ Is a simple algebraic extension and $f(x)=x_{n}+a^{n-1}x_{n-1}+...+a_{1}x+a_{0}$ ${\ displaystyle f (x) = x_ {n} + a ^ {n-1} x_ {n-1} + ... + a_ {1} x + a_ {0}}$ Is the minimal polynomial α, then ${\text{Tr}}_{K}^{E}(\alpha )=-a_{n-1}$ ${\ displaystyle {\ text {Tr}} _ {K} ^ {E} (\ alpha) = - a_ {n-1}}$

Expression of a trace through automorphisms E over K

Let σ ₁ , σ ₂ ... σ _m be all automorphisms of E leaving the elements of K fixed. If E is separable, then m is equal to the degree of [E: K] = n . Then for the trace there is the following expression:

${\text{Tr}}_{K}^{E}(\alpha )=\sigma _{1}(\alpha )+\sigma _{1}(\alpha )+\ldots +\sigma _{m}(\alpha )$ ${\ displaystyle {\ text {Tr}} _ {K} ^ {E} (\ alpha) = \ sigma _ {1} (\ alpha) + \ sigma _ {1} (\ alpha) + \ ldots + \ sigma _ {m} (\ alpha)}$

If E is inseparable then m ≠ n , but n is a multiple of m , and the quotient is a power of characteristic p: n = p ⁱ m .

Then ${\text{Tr}}_{K}^{E}(\alpha )=(\sigma _{1}(\alpha )+\sigma _{1}(\alpha )+\ldots +\sigma _{m}(\alpha ))^{m/n}$ ${\ displaystyle {\ text {Tr}} _ {K} ^ {E} (\ alpha) = (\ sigma _ {1} (\ alpha) + \ sigma _ {1} (\ alpha) + \ ldots + \ sigma _ {m} (\ alpha)) ^ {m / n}}$

Example

Let K be the field of real numbers , and E the field of complex numbers . Then the trace of the number $a+bi$ ${\ displaystyle a + bi}$ is equal to $2a$ ${\ displaystyle 2a}$ . The trace of a complex number can be calculated by the formula ${\text{Tr}}\;z=z+{\bar {z}}$ ${\ displaystyle {\ text {Tr}} \; z = z + {\ bar {z}}}$ , and this is in good agreement with the fact that complex conjugation is the only automorphism of the field of complex numbers.

Literature

Van der Waerden B. L. Algebra-M :, Science, 1975
Zarissky O. , Samuel P. Commutative Algebra Vol. 1-M :, IL, 1963
Leng S. Algebra-M :, World, 1967