Linked Questions

73
votes
7answers
16k views

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not ...
15
votes
2answers
3k views

Quadratic extensions in characteristic $2$

I recently saw in class that the degree $2$ extensions of a field of characteristic $\neq 2$ are given by square roots of non-squares in the base field. I wonder what happens in the case of ...
8
votes
4answers
2k views

Explicit examples of infinitely many irreducible polynomials in k[x]

My question is the following. Is it possible to give examples of infinitely many irreducible polynomials in a polynomial ring $k[x]$ with $k$ a field? I'm interested in this because I'm ...
5
votes
1answer
1k views

Quadratic equations that are unsolvable in any successive quadratic extensions of a field of characteristic 2

Show that for a field $L$ of characteristic $2$ there exist quadratic equations which cannot be solved by adjoining square roots of elements in the field $L$. In $\mathbb{Z_2}$ adjoining all square ...
3
votes
1answer
2k views

So-called Artin-Schreier Extension

Let $F$ be a field of characteristic $p$. Let $K$ be a cyclic extension of $F$ of degree $p$. Prove that $K=F(\alpha)$ where $\alpha$ is a root of the polynomial $p(x) = x^{p} - x - a$ for $a \in \...
1
vote
1answer
442 views

Radical extension and algebraic solution of an irreducible polynomial

Suppose that $k$ is a field with characteristic equal to zero, that $P \in k[X]$ is an irreducible polynomial and that $\alpha$ is a root of $P$ in an algebraic closure $\overline{k}$. Suppose also ...
2
votes
2answers
244 views

How to understand the Artin-Schreier correspondence?

Let $K$ be a field of characteristic $p > 0$. Then it is due to Artin and Schreier that the assignment $$c \in K \mapsto \text{Splitting field } L_c \text{ of } X^p-X+c$$ induces a bijection ...
2
votes
1answer
417 views

Artin-Schreier Theorem and Galois Extensions

This is what I know of the Artin-Schreier Theorem for field extensions: Let $K< L$ be a proper Galois extension such that $L$ has a prime characteristic $p\neq 0$ and $|\mathrm{Aut}(L/K)|=p$ then ...
0
votes
0answers
263 views

Showing a field extension is not solvable

I have a Galois field extension $E/F$ of degree $p$ and $F$ has characteristic $p$ and contains all the roots of unity. I've trying to show that $E/F$ is not a solvable extension. My main issue is ...
0
votes
0answers
250 views

Galois Group of $x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$

Question is to find Galois group of $f(x)=x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$ What i have done so far is : I could see that $f(x)$ is Irreducible and separable....
0
votes
1answer
70 views

is $x^2+x+1 \in \mathbb{F}_2$ solvable by radicals?

Is $x^2+x+1 \in \mathbb{F}_2$ solvable by radicals? First of all, this polynomial is irreducible in $\mathbb{F}_2$. It is also separable since f'(x)=1. But the zeros of f(x) are $(-1)^{2/3}, - (-1)^...
1
vote
1answer
54 views

Galois group of a Schreier solvable polynomial is solvable

I came across this problem which I couldn't solve thus far: Definition Let $F_0$ be a field with $char(F)=p>0$. A polynomial $f \in F_0[X]$ is said to be Schreier solvable if there exists a ...
3
votes
0answers
72 views

Finite separable normal extension has Galois abelian group

Prove that finite separable normal extension $\mathbb{F}$ of field $\mathbb{k}$, $\operatorname{char}(\mathbb{k})=p > 0$ has Galois abelian group $\operatorname{Gal}(\mathbb{F}/\mathbb{k})$ with ...
0
votes
0answers
61 views

Splitting field of $x^p − x + t$ not solvable over F

Given $p$ is a prime, $k$ is an algebraically closed field of characteristic $p$. and $F = k(t)$, where $t$ is a variable, let $L$ be the splitting field of $x^p − x + t$ over $F$. Then it can be ...
1
vote
0answers
20 views

Why p-cyclic extension iff $p^m$-cyclic extension $\forall m$

A theorem is stated as follows. For a field $F$ of characteristic $p$, $F$ has a $p$-cyclic extension if and only if for every positive integer $m$, $F$ has a $p^m$-cyclic extension. I wonder if ...