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3
votes
2answers
328 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 ...
2
votes
1answer
365 views

Hurewicz theorem related to Galois group (or Tannakian categories)?

Is there a proof of the Hurewicz theorem $\pi_1(X)^{ab} = H_1(X, \mathbf Z)$ ($X$ a connected topological space) expressing $\pi_1(X)$ as the "Galois" group of $X$, i.e., group of deck transformations ...
4
votes
2answers
1k views

Image of norm map for local field

Let $F$ be a finite extension of $Q_2$ (2-adic field) or $F_2((x))$ (function field). Let $E/F$ be a separable extension of degree $2$. What is the image of the norm map $N_{E/F}$? In particular - ...
7
votes
3answers
1k views

Infinite simple Galois groups

Conjecturally, every finite group is the Galois group of some extension of the rationals. This question made me wonder what is known about infinite simple groups occurring as Galois groups. What ...
9
votes
2answers
425 views

Which polynomials arise as formulas for a conjugate

For any integer $r \geq 2$, et $V_r$ be the set of polynomials $Q \in {\mathbb Q}[X]$ of degree $r-1$ such that there is an algebraic number $\alpha$ of degree $r$ , such that $Q(\alpha)$ is a ...
7
votes
1answer
352 views

Is H^2(W_p,C^times) well-known?

Let $W_p$ be the Weil group of $\mathbf{Q}_p$. What is the Galois cohomology group $H^2(W_p,\mathbf{C}^{\times} )$ (with trivial action)? Is it zero, or something huge and complicated? (This group ...
7
votes
2answers
2k views

What are Galois Categories used for?

Galois categories are introduced (for the first time?) in SGA1, but here's an English introduction that's available online: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Lynn.pdf It ...
5
votes
2answers
374 views

What is an example of a regular realization of $C_5$ over $\mathbb{Q}(x)$?

It's known that all abelian groups are regularly realizable over $\mathbb{Q}(x)$, but it occurred to me that I don't even have an example of a cyclic regular extension of $\mathbb{Q}(x)$ handy. So: ...
13
votes
0answers
453 views

Noether-Deuring for injections and surjections?

Noether-Deuring theorem (not in the strongest form, but in the one I usually need): Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over ...
2
votes
0answers
442 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 $L$ ...
5
votes
1answer
297 views

How many algebraic integers exist with degree $\leq k$ and bounds on the modulus of all Galois conjugates?

The precise question is the following: Question: Can one reasonably bound the number of algebraic integers $\alpha$ of degree at most $k$ - that means there exists a monic integer polynomial $p$ ...
4
votes
2answers
889 views

Cyclotomic extensions with split Galois group

$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q}$ Consider the set of all Galois extensions $E/\Q(\zeta_n)$ of a given cyclotomic field $\Q(\zeta_n)$ such that $$ \Gal(E/\Q) ...
2
votes
0answers
353 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 ...
3
votes
1answer
237 views

Are all solvable groups *regularly* realizable over Q(x)?

It is known for Hilbertian fields that all groups that are abelian, solvable, $A_n$ or $S_n$ are realizable over them. $\mathbb{Q}(x)$ is one such field, but it's not obvious that the extensions that ...
2
votes
3answers
2k views

What (permutation) groups can occur as galois groups of irreducible polynomials of degree n

I think the answers for the first few degrees ($n$) are: $n=2$, $S_2$ $n=3$, $S_3,A_3$ $n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group) $n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ...
18
votes
2answers
2k views

How to solve a quadratic equation in characteristic 2 ?

What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2? As pointed in the comments, it can be reduced to $X^2+X+c=0$ with $c\neq 0$. ...
5
votes
1answer
542 views

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 ...
20
votes
4answers
1k views

$A_5$-extension of number fields unramified everywhere

So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
8
votes
1answer
477 views

When are the intermediate fields totally ordered?

The groups whose subgroups are totally ordered by inclusion are easy to classify; they are subgroups of $\mathbb{Z}/p^{\infty} = \text{colim } \mathbb{Z}/p^k$ for some prime $p$, and thus ...
8
votes
3answers
437 views

Congruences mod primes in Galois extensions

I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields ...
4
votes
5answers
1k views

Why is every quadratic subfield of a Galois extension of the rationals with the quaternions as Galois group real?

Suppose that L is a field extension of the rationals with Galois group the quaternions Q={1,-1,i,-i,j,-j,k,-k}. Furthermore assume that L contains a quadratic subfield K. I have learned from this link ...
2
votes
1answer
208 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 ...
1
vote
0answers
293 views

Splitting of prime ideals in non-Dedekind domains?

This is a follow-up to this question. So that you don't have to flick back and forwards I'll briefly summarize: My original question was on how to prove that a polynomial $g(x)$ obtained from ...
8
votes
2answers
1k views

When is sin(r \pi) expressible in radicals for r rational?

Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary. As the question suggests, I would like to know when ...
11
votes
1answer
1k views

History of the Normal Basis Theorem

The Normal Basis Theorem: If $E/F$ is a finite Galois extension, then there exists $a \in E$ such that the orbit of $a$ under the action of $\mathrm{Gal}(E/F)$ is a basis for $E$ as a vector space ...
11
votes
2answers
444 views

Realizing D_8 as a Galois group over C(x) with prescribed decomposition groups

Coming up with examples of $D_8$-covers of $\mathbb{C}(x)$ is easy. For example: $Quot(\mathbb{C}(x)[y,z]/(y^2=x(x-7), z^4=(y+\sqrt{-6})^2(y-\sqrt{-6})^2(y+\sqrt{-10})(-y+\sqrt{-10})^3))$ defines a ...
12
votes
2answers
821 views

Proof of the result that the Galois group of a specialization is a subgroup of the original group?

I have been using the following result: Given a polynomial $f(x,t)$ of degree $n$ in $\mathbb{Q}[x,t]$, if a rational specialization of $t$ results in a separable polynomial $g(x)$ of the same ...
28
votes
8answers
7k views

Is Galois theory necessary (in a basic graduate algebra course)?

By definition, a basic graduate algebra course in a U.S. (or similar) university with a Ph.D. program in mathematics lasts part or all of an academic year and is taken by first (sometimes second) ...
3
votes
1answer
684 views

How do I visualize finite covers of curves over non-algebraically closed fields?

If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...
0
votes
1answer
917 views

Normal subgroups of the Galois group

I am trying to teach myself Galois theory. Is it true that every for a field extension K->L, that every normal subgroup of Gal(L:K) is of the form Gal(L:M) for some intermediate field M, ie K->M->L?
10
votes
5answers
3k views

Grothendieck's Galois Theory today

I have recently become aware of, and started to study in my free time (abundant in these summer months) Grothendieck's Galois Theory (GGT), as formulated in SGA 1 and later by Grothendieck's ...
4
votes
2answers
844 views

Galois theory of endomorphism rings of irreducible representations

Let $k$ be a field which I don't suppose to be algebraically closed. Then, the endomorphism ring of any irreducible representation of a finite group over $k$ is a division ring. What is known about ...
12
votes
5answers
3k 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 ...
5
votes
1answer
647 views

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} = ...
2
votes
2answers
714 views

algebraic numbers of degree 3 and 6, whose sum has degree 12

This question is related to Degree of sum of algebraic numbers. Forgive me if this is a dumb question, but are there two algebraic numbers $a$ and $b$ of degree $3$ and $6$ respectively, such that the ...
16
votes
1answer
842 views

Galois Group as a Sheaf

I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois ...
5
votes
2answers
2k views

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 ...
2
votes
1answer
408 views

Probability of an extension being normal

Let $P$ be the probability that an elliptic curve with a rational point has an infinite number of rational points. From what I understand, the value of P is unknown. This got me thinking about a ...
7
votes
2answers
829 views

Why do generic polynomials work in reality?

I understand that a generic $G$-polynomial $f(t_1,...,t_n)[X]$ over field $k$ has Galois group $G$ over $k(t_1,...,t_n)$. And basically any $G$ extension of $k$ should be generated by a realization of ...
6
votes
3answers
1k views

Computing only the order of Galois group (not the group itself).

My question is related to this one: Computing the Galois group of a polynomial. I was wondering if there is a faster algorithm just to compute the order of the group rather than the group itself. ...
11
votes
0answers
839 views

Galois theory timeline (II)

This question is a sequel. I structured the previous one around Emil Artin's classic treatment of Galois theory from the 1940s, though making clear some reservations of my own about whether Artin ...
1
vote
1answer
323 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 ...
3
votes
1answer
301 views

Is the (regular) inverse Galois problem known for the field C(x,y)?

I'd be surprised if somebody proved the inverse Galois problem for $\mathbb{C}(x,y)$, but I wanted to make sure.
42
votes
4answers
3k views

Degree of sum of algebraic numbers

This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer. Let $a$ and $b$ be algebraic numbers, with respective degrees ...
15
votes
2answers
2k views

Galois theory timeline

A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective ...
6
votes
3answers
3k views

What was Galois theory like before Emil Artin?

I read that the primitive element theorem for fields was fundamental in expositions of Galois theory before Emil Artin reformulated the subject. What are the differences between pre and post-Artin ...
2
votes
1answer
367 views

what is the first Galois cohomology group of the Galois module End(T_l(A)) for some abelian variety A over a finite field k and l some prime number different from the characteristic of the base field?

According to Serre's book 'Galois cohomology', Galois chomology group are always torsion, but it seems to me that H^1(k, End_{Z_l}(T_l(A)))=coker(Frob-1) on End_{Z_l}(T_l(A)), which has the same Z_l ...
3
votes
4answers
2k views

Cyclic extensions

Hi. Are there nice/simple examples of cyclic extensions $L/K$ (that is, Galois extensions with cyclic Galois group) for which $L$ cannot be written as $K(a)$ with $a^n\in K$? Thanks.
7
votes
1answer
849 views

Maximal extension almost everywhere unramified and totally split at one place

Fix a finite set of primes $S$ and an additional prime $p$. Let $K$ be the maximal extension of $\mathbb{Q}$ that is unramified outside $S$ and $\infty$ and totally split at $p$. Is the extension $K$ ...
4
votes
3answers
701 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 ...