All Questions
Tagged with ac.commutative-algebra galois-theory
61 questions
2
votes
1
answer
186
views
Behaviour of Primes under Regular Coefficient Extensions
Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x_{1},\cdots,x_{n}]\hookrightarrow K'[x_{1},\cdots,x_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring ...
- 125
2
votes
0
answers
1k
views
Decomposition group vs Galois group of completed extension for height > 1 primes
Assume
Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions.
Let $S$ be a finite $R$-algebra, $L$ its field of fractions.
$L/F$ a (finite) Galois extension
$S$ normal in $L$
...
- 21
2
votes
0
answers
384
views
What do you call an algebraic element with the property that the generated field extension is normal?
Let $L/K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Is there an established terminology for the property of $\alpha$ that $K(\alpha)/K$ is a normal field extension? Would you ...
- 2,399
6
votes
1
answer
951
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Is it possible to recover the degree of a field extension from a list of elements and the ground field?
I'm interested to know if there is anything known about recovering the degree of a field extension, $E/k$, given $E=k(\alpha_1,\ldots, \alpha_n)$ (here I'm assuming that the extension is of finite ...
- 1,049
2
votes
1
answer
286
views
Linear independence in the algebraic closure of $\mathbb{C}(z)$
Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.)
Define $w_i=(\...
- 454
16
votes
5
answers
5k
views
An advanced exposition of Galois theory
My knowledge of Galois theory is woefully inadequate. Thus, I'd be interested in an exposition that assumes little knowledge of Galois theory, but is advanced in other respects. For instance, it would ...
9
votes
1
answer
2k
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Symmetric polynomials theorem
Hello all, I would appreciate comments on the following question:
A main theorem of symmetric functions might be formulated: Let k be a field of char. 0. Then $k[x_1,...,x_n]^{S_n} = k[s_1,...,s_n]$,...
- 5,562
13
votes
2
answers
5k
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Primitive element theorem without building field extensions
Is there are nice way to prove the primitive element theorem without using field extensions?
The primitive element theorem says that if $x$ and $y$ are algebraic over $F$ and $y$ is separable over $F$...
- 1,439
1
vote
1
answer
349
views
Non-representability by a binary quadratic form
Let $k$ be an arbitrary field, $d\in k$, and $d$ is not a square in $k$.
Consider the binary quadratic form $f(x,y)=x^2-d y^2$
(it is the norm from $k(\sqrt{d})$ to $k$).
I am looking for a reference ...
- 14.1k
5
votes
3
answers
752
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Transformations of integer polynomials under combinations of their roots
I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
Preamble
We consider polynomials f &...
- 1,434
13
votes
2
answers
2k
views
Galois group of a product of polynomials
How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
- 2,782