All Questions
Tagged with galois-theory finite-fields
9 questions with no upvoted or accepted answers
17
votes
0
answers
750
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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
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 $\...
- 41.8k
3
votes
0
answers
285
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What is known about the prime-to-$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
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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
0
answers
70
views
Quadratic forms with the same roots over GF(2) for low rank problems
Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
- 63
1
vote
0
answers
88
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When does sum of algebraically independent polynomial become dependent?
Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a ...
- 341
1
vote
0
answers
139
views
Nature of polynomials of the form $x^n-a$ over finite fields
I state the following theorem from Serge Lang's Book- Algebra(3rd edition).
Theorem: Let $k$ be a field and $n$ an integer $\geq$ 2. Let $a\in k, a\neq 0$. Assume that for all primes $p$ such that $...
- 428
0
votes
0
answers
183
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Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 923
0
votes
0
answers
79
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Does "tensoring" with a fixed field preserve Galois extensions of finite fields?
Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}...
- 1,328