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Questions tagged [field-trace]

For questions concerning the trace of elements in field extensions.

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1answer
37 views

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 ...
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Trace of an algebraic integer is an integer?

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? ...
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solutions of gold APN functions using trace function

The Gold APN is defined as $F(x)=x^{2^{k}+1}$ in $GF(2^n)$, where $\gcd(k,n)=1$. The differential uniformity computed using $F(x)=F(x+a)=b$ as following: $x^{2^{k}+1} + (x+a)^{2^{k}+1}=b$ $x^{2^{k}+...
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1answer
35 views

Let $ f $ be an irreducible polynomial in $ \mathbb{F }_q [x] $, why $ f ^\frac{s}{deg (f)} $ has degree term $ s-1 $?

$ f $ is monic , $[E:\mathbb{F}_q]=s$ , $ E $ is an extension of $\mathbb{F}_q$ $deg(f)| s$ The book states that: $f(x)^\frac{s}{deg(f)} = x^s - c_{1}x^{s-1} ...- c_k$ I know $ f $ has deg (f) ...
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1answer
77 views

Trace form and totally real number fields

$\newcommand\Q{\mathbb{Q}}$ Let $K$ be a number field then there is a quadratic form over the $\Q$ vector space $K$ given by $$\tau: K\rightarrow \Q \qquad y\mapsto\mathrm{Tr}_{K/\Q}(y^2)$$ which is ...
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1answer
233 views

determining decimation ratio given characteristic polynomials of quotient rings $GF(2^n)$

Suppose I have $p_1(x), p_2(x) \in GF(2)[x]$ and fields $F_1 = GF(2)[x]/p_1(x), F_2 = GF(2)[x]/p_2(x)$ where both are isomorphic to $GF(2^n)$. I know that if $p_1(x) \neq p_2(x)$ then it is possible ...
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0answers
76 views

A statement about normal basis element , trace and characteristic of field

Let $F/K$ be a normal finite extension where $F=F_1 \times F_2$ for subfields $F_1,F_2$ of $F$ where $F_1\ne F$ . Suppose $w$ is a normal basis element in $F/K$ for any $w\in F$ for which $Tr_{F/F_2}(...