All Questions
Tagged with ac.commutative-algebra galois-theory
61 questions
2
votes
1
answer
159
views
Field extensions and completions at possibly infinite places
In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...
- 253
10
votes
1
answer
243
views
If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
- 452
3
votes
0
answers
114
views
English translation of Borel-Serre's "Théorèmes de finitude en cohomologie galoisienne"?
Is there an English translation of this text, or at least some English language paper that proves the same results?
I especially need a proof of the following fact which is in this paper: Say $k$ is a ...
- 31
1
vote
0
answers
88
views
When does sum of algebraically independent polynomial become dependent?
Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a ...
- 341
0
votes
0
answers
184
views
Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 923
8
votes
1
answer
639
views
A question on algebraic independence
Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...
- 341
1
vote
1
answer
100
views
Existence of a symmetric matrix satisfying certain irreducible conditions
Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
- 923
0
votes
1
answer
147
views
Existence of linear disjointness between an algebraic number field and $p$-cyclotomic field over $ \mathbb{Q}$
Suppose $ K $ be an algebraic number field and $ n $ be an even integer. Is it possible to find atleast one $ p $ such that $ p\equiv 1( \text{mod}~ n)$ and $ \mathbb{Q}(\eta_p) $ is linearly disjoint ...
- 923
6
votes
1
answer
245
views
If degree $N$ polynomials always have a root, does there still exist an irreducible polynomial of degree $N+1$?
Given that $F$ is a field, let $F_n$ be the completion of $F$ with respect to roots of degree $n$ polynomials. For example this would make $\mathbb{Q}_2$ the field of (ruler and compass) constructible ...
- 700
3
votes
1
answer
1k
views
irreducibility of the polynomial $ x^4 +1 $
Let $K$ be a field. We consider the polynomial $f(x) = x^4 + 1$. It is known that $f(x)$ is irreducible over $\mathbb{Q}$ but reducible over any finite field. Thus $ f(x)$ is reducible over any field $...
- 923
6
votes
1
answer
533
views
Construction of a symmetric polynomial in the roots that acts like the discriminant
The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...
6
votes
2
answers
329
views
Algebraic numbers which prescribed degree which does not belong to some fields
In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...
- 515
10
votes
0
answers
452
views
What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?
Consider the following statement:
If $K\subseteq L$ is a Galois extension of fields with Galois group $G$ and $x \in \mathbb{P}^n(L)$ is such that $\sigma(x)=x$ for all $\sigma\in G$, then $x \in \...
- 32.5k
4
votes
0
answers
212
views
Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
2
votes
0
answers
312
views
Degree $8$ cyclic extension over $\mathbb{Q}$
Actually I am interested in degree $ 8 $ cyclic extension over $ \mathbb{Q} $. Let $ L $ be such extension. At first I was thinking to take basis as normal basis, as we can determine the galois group ...
- 923
0
votes
0
answers
112
views
Existence of a subspace of having no isotropic 2-plane
Let $V$ be a vector space of dimension $n$ over the field $\mathbb {Q} $. A subspace $W$ is isotropic for a skew-bilinear form $\alpha$ on $V$ if $\alpha(x,y) = 0$ for all $x,y \in W$.
More ...
- 923
5
votes
2
answers
243
views
Restricting maps between strict henselisations
$\require{AMScd}$I am currently thinking about (strict) henselisations but I don't know too much literature about the topic. So I am wondering if there is a natural way to restrict maps between strict ...
- 153
1
vote
0
answers
64
views
Resolvent is minimal polynomial for universal splitting algebra
Given a degree $n$ monic $f\in A[x]$ write $\mathrm{Split}_Af$ for its universal splitting algebra, constructed by taking the quotient of $A[x_1,\dots ,x_n]$ by the Vieta formulas. This is the initial ...
- 10.5k
1
vote
2
answers
368
views
How to show an invariant subfield of rational function field $\mathbb{Q}(x)$ under a certain group action is actually a simple extension? [duplicate]
Let $K=\mathbb{Q}(x)$ be the rational functions in one variable $x$ and let the automorphisms $\phi,\psi$ of $K$ be defined as $\phi(x)=-\frac{1}{x+1}$ and $\psi(x)=\frac{1}{x}$.
Let $G$ be the group ...
user178596
2
votes
0
answers
221
views
Field whose absolute Galois group is $\mathbb{Z}_p$
Let $L$ be a perfect field. Assume there is an algebraic closure $L\subset \overline{L}$ and a prime $p$ such that $\mathrm{Gal}(\overline{L}/L)\cong \mathbb{Z}_p$ as topological groups.
Is there a ...
- 53
0
votes
2
answers
757
views
Splitting field of an intermediate field
Consider the following 'wrong' question.
Let $f(x) \in F[x]$ be an irreducible polynomial in a polynomial ring of a field $F$. Let $L$ be the splitting field of $f(x)$ over $F$. Assume that $L$ is a ...
- 103
2
votes
1
answer
166
views
On a special type of subring of $\mathbb C[x_0,...,x_{q-1}]$
Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let
$$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...
- 1,209
2
votes
0
answers
27
views
Does the $G$-norm coincide with the ordinary norm for "quasi-$G$-Galois" extensions
Let $S$ be a commutative ring, let $G$ be a finite group acting on $S$ via automorphisms (not necessarily faithfully), and
let $R$ be a subring of $S$ consisting of elements fixed $G$.
The extension $...
- 2,928
18
votes
1
answer
987
views
Why does $E\otimes_KE\cong EG$ imply that Galois theory works?
This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the Galois group of $E/K$. $E[G]$ is the group ring formed by finite group $G$.
"It is ...
- 299
6
votes
2
answers
815
views
Why is $K_{\upsilon}|K$ separable for a global field $K$?
I asked this question on math.stackexchange but maybe it fits here better. If not, I apologize in advance and will remove the question.
Let $K$ be a global field and $\upsilon$ a prime of $K$. Then ...
- 111
2
votes
0
answers
171
views
residue fields of smooth $\mathbf{Q}$-algebras
Let $A$ be a $\mathbf{Q}$-algebra. We say $A$ is "residually abelian", if there exists a maximal ideal $\mathfrak{m}$ of $A$ whose residue field $\kappa(\mathfrak{m})$ is a Kummer extension of an ...
user92332
2
votes
1
answer
485
views
Extending commuting automorphisms of a field
I have a field $L$ and two field automorphisms $f, g: L \to L$ such that $f \circ g = g \circ f$. Under what conditions can they be extended to automorphisms of an extension $K \mid L$ that still ...
- 651
1
vote
0
answers
128
views
Nielsen--Schreier for fields
Is it true that a subextension of a purely transcendental extension is itself purely transcendental?
In symbols, suppose we have field extensions $M/L/K$, with $M/K$ purely transcendental. Must $L/K$ ...
- 9,720
1
vote
0
answers
36
views
Quadratic suborders of an imprimitive quartic order
Let $Q$ be an irreducible quartic order; that is, $Q$ is a subring of the ring of integers $\mathcal{O}_K$ in a quartic extension $K$ over $\mathbb{Q}$ such that the fraction field of $Q$ is equal to $...
- 26.9k
28
votes
4
answers
4k
views
What are traces?
Let $A$ be a Noetherian commutative ring and Let $A\rightarrow B$ be a finite flat homomorphism of rings. We can thus form the so called "trace" $\mathrm{Tr_{B/A}}:B\rightarrow A$, which is a ...
- 953
2
votes
0
answers
192
views
Existence of "splitting objects" and algebraic closures
Definition 0. Let $(\mathbf{C},U)$ denote a concrete category. Let "splits" be a desirable property that pointed $\mathbf{C}$-objects may or may not satisfy. Let $(X,x)$ be a pointed $\mathbf{C}$-...
- 3,793
11
votes
1
answer
2k
views
Galois theory for products of fields (aka finite etale extensions)
Let $F$ be a field. By a Galois algebra over $F$ I mean a finite etale extension, that is, a product $K = K_1 \times \cdots \times K_r$ of finite (separable) field extensions, of total degree $[K : F]...
- 980
5
votes
1
answer
339
views
Can completely multiplicative functions be extended to $\overline{\mathbb{Q}}$ or further?
I'm looking for a subject of study that handles the following question. I'm not the most familiar with algebra; I have a strong working knowledge and that's about it, but I've been considering ...
user78249
1
vote
0
answers
81
views
Geometry of componentially locally strongly separable algebras
Janelidze's categorical Galois theory yields, for nice adjunctions, a good notion of covering morphism.
The category of finitely affine schemes admits such an adjunction into the category of ...
- 10.5k
20
votes
2
answers
1k
views
Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups
A classic result says the automorphism group of $\mathbb{R}$ (over $\mathbb{Q}$) is trivial. The proof is simple: every automorphism preserves squares, and hence fixes the positive reals, so it must ...
- 201
-2
votes
1
answer
151
views
Quadratic extension and prime ideals
Let $B/A$ be a quadratic Galois extension between local domains. Define ${\mathrm{Gal}}(B/A) = \{e,\sigma\}$.
Choose two prime ideals ${\frak P}_1, {\frak P}_2$ of $B$ such that ${\frak P}_2 = {\...
- 781
0
votes
1
answer
232
views
Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$
I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals.
The ...
- 591
12
votes
0
answers
3k
views
Algebraic Closure of the field of rational functions
Using the theorem of Puiseux, one concludes that the algebraic closure of $\mathbb C(X)$ is the set of algebraic elements (over $\mathbb C(X)$) of the algebraic closure of $\mathbb C((X))$, which is $\...
- 1,891
3
votes
2
answers
243
views
Minimal fields of isomorphism for varieties
Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
- 11.3k
6
votes
1
answer
760
views
Complexity of computing the Galois group
There has been some discussion of computing the Galois group of a polynomial over the integers, but I can't seem to find any results, or even a question of what the complexity of this might be. For ...
- 96.4k
2
votes
0
answers
60
views
lift isomorphic in a sufficiently thick fiber
Let $P$ and $P'$ two polynomials in $k[[\pi]][t]$ for an algebraically closed field $k$ and let $A=k[[\pi]]$.
We consider $ X'=Spec (A[t]/(P'))$ and $X=Spec (A[t]/(P)$.
Let $d=val(\Delta(P))$ where $\...
- 3,472
2
votes
0
answers
173
views
On Artin-Hironaka lemma and Galois theory
Let $A=k[[t]]$ Let $B$ a flat $A$-finite algebra which is etale and Galois at the generic point.
Then by Artin lemma 3.12 (ii) in his IHES paper on approximation, we know that there exists an integer ...
- 3,472
0
votes
0
answers
245
views
Notation Problem, Fixed Rings and Fields
I am trying to make sense of the notation and certain sets in two articles by Annick Valibouze whose results I'm using for my bachelor's thesis, I hope it's relevant enough to merit an answer.
In one ...
- 119
10
votes
2
answers
2k
views
why are subextensions of Galois extensions also Galois?
Generally a Galois extension is defined to be an algebraic extension that is also normal & separable. It is then shown that in the sequence of field extensions $L|M|K$ if $L|K$ is Galois then $L|M$...
- 2,083
8
votes
1
answer
4k
views
Example of an algebraic number of degree 4 that is not constructible
The number $b:=\frac{\sqrt{2a}+\sqrt{4\sqrt{a^2-3}-2a}}{2}$ with
$a:=\frac{\sqrt[3]{18+2\cdot\sqrt{65}}}{2}+\frac{2}{\sqrt[3]{18+2\cdot\sqrt{65}}}$ is a root of the irreducible polynomial $x^4-6x+...
- 143
2
votes
1
answer
652
views
Searching for polynomials with squarefree discriminant
In connection with the Galois theoretic results surrounding the irreducibility of $f(x)= x^{N}-x-1$ over $\mathbb{Q}$, I've been trying to prove for a while that the discriminant of $f$ is actually ...
- 625
11
votes
2
answers
3k
views
Is there an alternative formula for solving cubic equations?
It is known that Cardano's formula for solving cubic equations is not good in the case of positive discriminants. In this case it expresses the solution through cubic roots of complex numbers. ...
- 111
15
votes
3
answers
4k
views
Elementary Luroth theorem proof?
Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...
- 1,422
3
votes
0
answers
603
views
Norms in Galois extensions
Let $k$ be a field of characteristic 0, and $\overline k$ be a fixed algebraic closure of $k$.
Let $k\subset F\subset E$ be a tower of finite Galois extensions in $\overline k$,
where both $\mathrm{...
- 14.2k
24
votes
4
answers
4k
views
Is there a Galois correspondence for ring extensions?
Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to ...
- 1,415