# Questions tagged [field-trace]

For questions concerning the trace of elements in field extensions.

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### Trace map from $\Bbb Q\Bbb(\zeta_r)$ to $\Bbb Q\Bbb.$

I am interested in knowing how to get values of $Tr_r(a)$. Where $Tr_r$ is trace map from $\Bbb Q\Bbb(\zeta_r)$ to $\mathbb Q\mathbb,$ and $\zeta_r$ are some specific complex roots of unit. Not really ...
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### Number of solutions of system of linear equations over finite field

Suppose $F=GF(2^8)$. Let $u_1,u_2\in K=GF(2^4)$ be linearly independent elements. The functions $x\mapsto tr_n (u_ix)$ are linear functions from $F\to GF(2)$, $i=1,2$. Here $tr_n$ denotes the absolute ...
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### definition of the trace by conjugates and using the characteristic polynomial

The trace of an element $\alpha \in \Bbb F_{q^m}$ over $\Bbb F_q$ is defined to be the sum of all its conjugates, i.e. $$\mathrm{Tr}_{F/K}(\alpha)=\sum_{i=0}^{m-1}\alpha^{q^i}.$$ I have also seen it ...
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### Members of finite field with equal Norm and Trace

The setting of the problem I'm trying to solve is as follows: Let $p$ be a prime number and $k \geq 1$. How many pairs $x, y \in \mathcal F_p^k$ satisfy $\text{Tr}(y) = N(x)$? My first intuition ...
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### Let $K|F$ be a finite separable extension (algebraic), then show that $\operatorname{Tr}_{K|F} : K \to F$ is surjective.

Let $K|F$ be a finite separable extension (algebraic), then show that $\DeclareMathOperator{\Tr}{Tr}\Tr_{K|F} : K \to F$ is surjective. Note: $F$ is not assumed to be finite like here , so not a ...
Let $F$ be a number field and let $\alpha \in F$. If $\alpha \in \mathcal{O}_F$, then it is known that $N(\alpha) \in \mathbb{Z}$. I was wondering if something similar can be said about the trace? ...