# Tagged Questions

441 views

### Is Frobenius the only magical automorphism?

The Frobenius automorphism is special because the $p$-power map makes sense in any characteristic $p$ ring, which allows us to canonically extend the Galois-theoretic Frobenius to any such ring. I ...
60 views

### Characterizing Galois field extensions via tensor product

Let $K\subset L$ be a finite field extension of degree $n$. Prove that it is Galois if and only if $L\otimes_{K} L\simeq L^{n}$, as $L$-algebras, considering on $L \otimes_{K}L=L^{n}$ the left ...
46 views

### question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
31 views

63 views

### Integral Galois Extension (Serge Lang)

I have two questions about the proof of the following Proposition: Let $A$ be a ring, integrally closed in its quotient field $K$. Let $L$ be a finite Galois extension of $K$ with group $G$. Let $P$ ...
74 views

### A criterion for an extension to be Galois

This is an exercise given during my Commutative Algebra course. I reached to solve just the "if" arrow, but not the "only if". The question is: Let $F\subseteq L$ be a finite degree extension of ...
33 views

### Explicit display of the contraction of an ideal in polynomial ring extensions

$K$ is a finite Galois extension of $k$, $I=(f_1,\dots,f_m)$ an prime ideal in $K[X_1,\dots,X_n]$, then what is $I\cap k[X_1,\dots,X_n]$? Is it ...
387 views

### Is every rigid field perfect?

A field is rigid iff its automorphism group is trivial. A field $F$ is perfect iff all irreducibles in $F[x]$ are separable. Is every rigid field perfect?
74 views

### proving closure of a set of automorphisms in the Krull topology in the context of integral ring extensions

Let $A$ be an integrally closed integral domain with field of fractions $K$ and let $p \in Spec(A)$. Let $L/K$ be a Galois extension with group $G$ and $L'/K$ be a finite Galois subextension. Let $B$ ...
337 views

### Maximal ideals in polynomial rings over a field

Let $K$ be an algebraically closed field and let $k$ be a subfield of $K$ such that the field extension $K \mid k$ is algebraic. Let $B$ be the polynomial ring $K [x_1, \ldots, x_n]$ and let $A$ be ...
143 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 ...
102 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 ...
55 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 ...
94 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 ...
170 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 ...
225 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 ...
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 ...