All Questions
Tagged with galois-theory ra.rings-and-algebras
30 questions
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
1
answer
188
views
Is there a uniform family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ such that $f_p(x)\in \mathbb{Z}[x]$ is irreducible and irreducible mod $p$?
Let $p\in\mathbb{Z}$ be a positive prime number.
Is there a "uniform" family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ of degree two such that $f_p(x)\in \mathbb{Z}[x]$ is both irreducible ...
- 237
0
votes
0
answers
33
views
determinantal ideal of sum of Galois conjugate matrices
Given $n$ matrices $A_i \in \mathbb{Z}^{m\times m}$. I am interested in the ideal $I_d(A)$ generated by the $d\times d$-minors of $A = \sum_{i=1}^n x_iA_i \in \mathbb{Z}[x_1, \dots , x_n]$.
The matrix ...
- 61
0
votes
0
answers
57
views
E=F(E^p) implies extension is algebraic
Let $F$ be a field of characteristic $p$ and $E/F$ be a finitely generated field extension such that $E=F(E^p)$. Then show that $E/F$ is algebraic.
I have proven it in case $E$ is singly generated ...
- 145
1
vote
0
answers
124
views
$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?
We have the following identities:
$\sin(\frac{\pi}{1})=0$
$\sin(\frac{\pi}{2})=1$
$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$
$\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$
Lets start with a definition.
Rules ...
- 763
0
votes
1
answer
259
views
Does there exist a proper intermediate field between ℚ and ℚ̅ closed under taking nth roots? [closed]
Title says it all. Sorry my previous question was wrong; I see now; very stupid of me. So this is what I meant to ask. I am looking for a field extension of $\mathbb Q$, let's call it $K$, s.t. $K$ is ...
- 13
6
votes
0
answers
204
views
The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?
I asked this question over a year ago on Math.StackExchange but I didn't get an answer.
In his famous treatise On spirals, Archimedes used a spiral to square the circle and trisect an angle. There are ...
- 841
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
1
vote
3
answers
1k
views
What are the main open problems in Group Theory and Galois Theory? [closed]
I’m really interested in doing a PhD and the subjects I enjoyed the most were Group Theory and Galois Theory.
What are some open problems in these areas that would be suitable for a PhD? Is Galois ...
6
votes
1
answer
518
views
What is the Galois group of one ultrapower over another ultrapower?
Let $F$ be a field, let $E$ be a field extension of $F$, and let $U$ be an ultrafilter. Then my question is, what is the relationship between the Galois groups $Gal(\Pi_U E/\Pi_U F)$ and $Gal(E/F)$?
...
- 4,597
1
vote
1
answer
101
views
When is a infinite transcendence-degree rigid fields fixed by a finite extension?
A field is rigid if it has no nontrivial automorphisms. Let $F$ be a rigid field which has infinite transcendence degree over $\mathbb{Q}$, and let $E$ be a finite extension of $F$. Then my question ...
- 4,597
1
vote
1
answer
488
views
Polynomials for the alternating group $A_n$
It is my understanding that the polynomial $f_n(x)=x^n-1$ has the Galois Group $(\mathbb{Z}/n\mathbb{Z})^*$, group of units of order $\phi(n)$. In some sense, these are the "simplest" ...
2
votes
1
answer
342
views
Name and properties of this combination of group algebra and semidirect product?
Given a field $k$, a group $G$, and a homomorphism $\phi : G \to \mathrm {Aut} (k)$, we can define a ring $\widehat {k [G]}_\phi$ as follows: As an abelian group it is isomorphic to the group algebra $...
- 1,105
1
vote
1
answer
324
views
Galois group of an L-function
Let $ M $ be a class of L-functions such that whenever $ F $ and $ G $ belong to $ M $, then so do their product $ F.G $ and their tensor product $ F\otimes G $ defined by $ F\otimes G : s\...
- 6,974
3
votes
1
answer
213
views
Extension field $\mathbb{C}(t,u)$ over $\mathbb{C}(t^n,u^n)$
I'm teaching myself some mathematics, so post question here sometimes is my last resort to get an answer, i have already posted this question on Mathematics Stack Exchange But no one answers, and I ...
- 125
4
votes
1
answer
177
views
Hopf-Galois Structure Maps
A Hopf-Galois extension of commutative rings, as defined by Montgomery here, is a morphism of commutative rings $\phi:A\to B$ with a Hopf-algebra $H$ coacting on $B$ by a ring map $c:B\to B\otimes H$ ...
- 10.4k
3
votes
1
answer
180
views
Bibliography suggestion for Kummer theory
I already posted a question about a sum involving the degree of a Kummer extension.
Now I'm interested in a more specific fact about Kummer extensions.
From Hooley's paper "On Artin's conjecture", we ...
- 215
5
votes
0
answers
241
views
Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$
I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question
In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...
- 356
5
votes
2
answers
580
views
Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $Gal(T(f))$ $\simeq$ The commutator subgroup of $Gal(f)$?
I asked this question at MSE but I did not receive an answer. So I ask it at MO:
We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by $Gal(f)$...
- 356
2
votes
1
answer
541
views
What are some good references on the Galois theory, factorization, or minimality of differential equations?
Let $\lambda$ be a nonzero complex number and let $u(x)$ be some smooth function $\mathbb{R}\to\mathbb{C}$, not identically zero. I want to prove that if $u$ satisfies $$u'' + \lambda u'+ \lambda^2 u =...
- 123
3
votes
2
answers
243
views
Minimal fields of isomorphism for varieties
Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
- 11.3k
0
votes
2
answers
190
views
power sums are enough for rationality? [closed]
If I have k algebraic integers like a_1, ..., a_k such that the sum of their n-power are integer for n=1, ...m
can we deduce that a_1, ..., a_k are integers? how large m should be? (how many power ...
- 427
2
votes
1
answer
693
views
When does a polynomial split over Q?
If P(x) is a polynomial in Q[X], is there any iff theorem that states all the roots of P(x) are rational based on the coefficients?!
In another words, what could you impose on the coefficients to ...
- 427
6
votes
3
answers
1k
views
Solving $z^n=a+bi$ using only radicals of positive real numbers
Let $a+bi\in\mathbf{C}$ be a complex number with $a,b\in\mathbf{R}$. Then
it is easy to find exact solutions of $z^2=a+ib$. For example let $z=u+iv$. Then
$$
u^2+v^2=z\overline{z}=\sqrt{a^2+b^2}
$$
...
- 7,609
1
vote
1
answer
542
views
Is there a subfield $F$ of $\mathbb{R}$ such that $\mathbb{R}$ is a finite algebraic extension of $F$. [duplicate]
Possible Duplicate:
Examples of algebraic closures of finite index
The question is in the title.
I can prove that if such field $F$ exist then the extension $\mathbb{R}/F$ cannot be of degree $2$...
- 1,701
5
votes
3
answers
711
views
Octic family with Galois group of order 1344?
Does the octic,
$\tag{1} x^8+3x^7-15x^6-29x^5+79x^4+61x^3+29x+16 = nx^2$
for any constant n have Galois group of order 1344? Its discriminant D is a perfect square,
$D = (1728n^4-341901n^3-...
- 12.6k
2
votes
1
answer
292
views
nth-powers and degree n polynomials with coefficients in field extensions
Hi,
Suppose that $E/F$ is a Galois extension. If $P(X)\in E[X]$ is a (EDIT: monic) polynomial of
degree $n > 0$, such that $P(X)^n\in F[X]$, does it follow that $P(X)\in F[X]$?
Thanks
- 2,842
4
votes
2
answers
483
views
Rank of sum of Galois conjugates of a matrix
Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in \mathrm{...
- 1,810
5
votes
3
answers
1k
views
Splitting of a division algebra with an involution of second kind
Let $k$ be a field, $K/k$ a separable quadratic extension,
and $D/K$ a central division algebra of dimension $r^2$ over $K$
with an involution $\sigma$ of second kind
(i.e. $\sigma$ acts non-trivially ...
- 14.1k
43
votes
3
answers
7k
views
transcendental Galois theory
Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ of $K \setminus L$, ...
- 65.4k