All Questions
7 questions
17
votes
0
answers
750
views
Elements of finite fields with many powers of trace zero
Let $p$ be an odd prime number, $n>1$ be an integer, and $\mathrm{tr}$ be the trace map of the field extension $\mathrm{GF}(p^{2n})/\mathrm{GF}(p)$. For which pair $(p,n)$ does there exists $x\in\...
- 767
11
votes
3
answers
1k
views
Cubic polynomials over finite fields whose roots are quadratic residues or non-residues
For a cubic polynomial $f(x)=x^3+x^2+\frac{1}{4}x+c$ over $\mathbb{F}_q$, where $q$ is a odd prime power, I find that for a lot of $q$, there does not exist $c\in\mathbb{F}_q$ such that $f$ has three ...
user150240
9
votes
0
answers
261
views
Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?
$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...
- 41.9k
8
votes
1
answer
355
views
The distribution of certain Galois groups
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Let $G_p^+$ be the Galois group of the polynomial $f(x)-y$ over $\overline{\mathbb{F}}_p(y)$ and $G_p$ be the Galois group of the ...
- 7,193
3
votes
0
answers
285
views
What is known about the prime-to-$p$ etale fundamental group of $\mathbb{P}^1_{\mathbb{F}_p}$ minus $\mathbb{F}_p$-rational points?
Is it known to be (the prime-to-$p$ part of the profinite completion of) a finitely presentable group?
Is such a presentation known? Is there a guess for what it is? What is known about it?
- 2,071
2
votes
0
answers
124
views
Can K$_3$ of finite fields be related to Teichmüller cocycles?
This is sort of a blind shot, but...
For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$.
To simplify matters, let $R$ be a finite field $\mathbb ...
- 18.4k
1
vote
1
answer
323
views
Maximal separable extension of $\mathbb F_q((t))$
Let $K=\mathbb F_q((t))$. I want to prove that $K^{sep}$ is composite of $K^{sep}(p)$ and $K^{sep}(not \ p)$, where $K^{sep}(p)$ is maximal Galois extension of $K$ of exponent $p$, $K^{sep}(not \ p)$ ...
- 11