Questions tagged [field-trace]

For questions concerning the trace of elements in field extensions.

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47 views

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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1answer
28 views

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 ...
2
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1answer
97 views

Norm and trace of an element in a cyclotomic number field

Let $K$ be a number field of degree $n$ over $\mathbb{Q}$, and let $\alpha \in K$. There are $n$ distinct embeddings of $K$ into $\mathbb{C}$ -- and we will denote these by $\sigma_1, \sigma_2, ... , \...
2
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1answer
49 views

Trace Form on Product of Fields

I am studying algebraic number theory and I am having trouble understanding something. Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Suppose a prime $p$ does not ramify in $K$. Then ...
2
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1answer
69 views

Existence of orthogonal base for finite Galois extension over characteristic 2

Let $K$ be a field of characteristic $2$ and $L$ be a finite Galois extension of $K$. Considering the trace $Tr_{L/K}: L \to K$ and $L$ as a finite dimensional $K$-vectorspace we know, that $Tr_{L/K} \...
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0answers
16 views

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 ...
1
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1answer
72 views

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 ...
2
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1answer
163 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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2answers
459 views

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? ...
3
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0answers
76 views

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}+...
2
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1answer
41 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) ...
2
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1answer
138 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 ...
3
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1answer
252 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 ...
2
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0answers
89 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}(...