Tagged Questions

11
votes
1answer
73 views

Uniformly solvable families of polynomials

It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
2
votes
1answer
91 views

Making the fundamental theorem of Galois theory explicit

I encountered the present question when investigating that other recent question of mine. Let $x_1,x_2, \ldots, x_8$ be indeterminates. Let $s_1,s_2, \ldots s_n$ denote the elementary symmetric ...
2
votes
1answer
49 views

How many prime ideals are fixed by a given permutation?

Suppose $L$ is a finite Galois field extension of the rational number field $\mathbb{Q}$, and $B$ is the integral closure of $\mathbb{Z}$ in $L$. Let $\sigma$ be an element of the Galois group ...
2
votes
1answer
85 views

Technical question on integral ring extensions

Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a finite Galois extension of $K$ with group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $p$ be ...
2
votes
1answer
160 views

Integral Galois Extensions - Proposition 2.4 (Lang)

My question refers to the proof of Proposition 2.4, p. 341 in Lang's Algebra. Here is the context: Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a ...
6
votes
1answer
184 views

what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?

John Conway proved in his book, On Numbers and Games (ch6, theorem 49) that the set of all ordinals smaller than $\omega^{\omega^\omega}$ form a field of characteristic 2 that is isomorphic to ...
1
vote
1answer
77 views

When does the fraction field of a ring have a non-trivial Galois extension

I have read this previous question on existence of a non-trivial Galois extension. I was wondering about the following situation. Suppose, $R$ is a domain that is not a field. When does the fraction ...
9
votes
3answers
285 views

Galois ring extension

Is there an analogous theory to Galois extension of fields for commutative rings? In particular, what does it mean for a ring extension to be Galois? Thanks.