15 questions linked to/from Reference book for Artin-Schreier Theory
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 ...
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 ...
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 ...
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 ...
2k views

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 ...
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 ...
### 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 ...
### 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 ...