Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [field-trace]

For questions concerning the trace of elements in field extensions.

33 questions
3
votes
1 answer

Are there explicit formulas for traces and norms of $p$-adic fields with given ramification index?

I met the following "elementary fact" on Henniart-Bushnell's text Local Langlands Conjectures for $GL(2)$ (which they didn't include a proof): Let $F$ be a $p$-adic field and $E/F$ a finite extension. Then $E/F$ is tamely ramified if and only if…
youknowwho
  • 1,553
3
votes
2 answers

Subfield generated by traces of some elements in finite fields

I'm doing exercises on Bonnafe's beautiful book Representations of $SL_2(\mathbb{F}_q)$. The first exercise is: 1.1. Let $k$ be the subfield of $\mathbb{F}_{q}$ generated by $\{ \operatorname{Tr}_{2}(\xi):\xi\in \mathbb{F}_{q^2} ~,\xi^{q+1}=1\}$.…
youknowwho
  • 1,553
3
votes
1 answer

If $\operatorname{tr}(ab)=0$, then $f(x)=x+a\operatorname{tr}(bx)$ is a permutation

Let $L/K$ be finite fields with elements $a,b\in L$ , with $\operatorname{tr}$ being the trace map and $f:L\to L$ given by $f(x)=x+a\operatorname{tr}(bx)$. Then,is it true that, $\forall x\in L$, if and only if $\operatorname{tr}(ab)=0$, then…
vidyarthi
  • 7,315
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1 answer

Finite field trace property, what is the analog for characteristic 0

Background: Given an extension of finite fields $\mathbb{F}_{q^r} / \mathbb{F}_q$ where $q$ is a prime power, the field trace with respect to this extension is given by $$\text{tr}_{\mathbb{F}_{q^r} / \mathbb{F}_q}(\alpha) =…
Aidan W. Murphy
  • 1,436
3
votes
1 answer

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 duplicate! My attempt: The finite case is done …
user422112
3
votes
1 answer

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: $$x^{2^{k}+1} + (x+a)^{2^{k}+1}=b$$ $$x^{2^{k}+1} + (x+a)^{2^{k}}(x+a)=b$$ $$x^{2^{k}+1} +…
hardyrama
  • 217
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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 also known as the trace-form of $K$. As far as I…
quantum
  • 1,779
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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 to show that $\operatorname{Tr}_{F_1}(x^k) =…
Jason S
  • 3,179
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1 answer

Whether a $n-1$ dimensional subspace of $\mathbb{F}_{q^n}$ can be written as a multiple of the kernel of the trace map

Let $q$ be a prime power and $n$ be a positive integer. Let $U$ be a $\mathbb{F}_q$-vector subspace of the finite field $\mathbb{F}_{q^n}$ of dimension $n-1$. Let $V$ be the kernel of the trace map $Tr: \mathbb{F}_{q^n} \rightarrow \mathbb{F}_q$. It…
jimm
  • 1,065
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1 answer

Trace on Finite fields

Let $q$ be a prime power. Denote by $\mathbb{F}_{q^m}$ the finite field with $q^m$ elements for any positive integer $m$. Fix a positive integer $r$, and fix $\alpha \in\mathbb{F}_{q^r} $. Prove the trace $T_{\mathbb{F}_{q^r}/…
Korn
  • 1,618
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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 $\sigma_1, \sigma_2, ... , \sigma_n$. The norm of $\alpha$ is given by…
michiganbiker898
  • 1,573
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1 answer

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 we can write $p\mathcal{O}_K =…
sqrt-3299
  • 297
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1 answer

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 $Tr_{L/K} \neq 0$ hence we get a non-degenerate…
ctst
  • 1,442
2
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2 answers

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? I know that the trace of any element in $F$ is in…
yktd
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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) distinct roots in $ E $, but I do not see how to have a…
user489941
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