The galois-theory tag has no wiki summary.

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### Where is a good place to start learning about the Grothendieck-Teichmuller group?

I've had a desire to get some sort of handle on the Grothendeick-Teichmuller group for years now, but I've always felt that I could never find a source that was introductory and readable.
The obvious ...

**5**

votes

**1**answer

368 views

### Given a branched cover with branch cycle description $(g_1,…,g_r)$, does $g_i$ generate some decomposition group?

Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so ...

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votes

**1**answer

283 views

### Finite topological generation of Galois groups

Let $F/\mathbf{Q}$ be an extension of finite degree, and let $S$ be a finite set of places of $F$. Let $F_S/F$ be the maximal extension unramified outside $S$; what is the most natural way to see ...

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1k 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 ...

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1k views

### Finitely generated Galois groups

It is well-known that for a given natural number $n$ there is only finite number of extensions of $\mathbb Q_p$ of degree $n$. This result appears in many introductory books on algebraic number ...

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votes

**2**answers

557 views

### Centralizers of elements in free profinite groups

I believe, although I can't say that I've given a rigorous proof, that for a free group $F_r$, and an element of it $a$, $C_{F_r}(\langle a \rangle)=$ the group generated by the elements $b \in F_r$ ...

**12**

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957 views

### Using higher-order Bring radicals to solve arbitrary polynomials

It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula that only involves addition, subtraction, ...

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votes

**1**answer

369 views

### What is the size of the smallest rigid extension field of the complex numbers?

Suppose we consider a rigid extension field $F$, i.e., $\text{Aut}(F) = 1$ over the complex numbers $\mathbb C$. What is the minimal cardinality of $F$? In particular it should hold that in this case ...

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173 views

### Decomposing anticyclotomic characters

Suppose $K/\mathbf{Q}$ is an imaginary quadratic field and $\chi$ is a finite-order character of $G_K=\mathrm{Gal}(\overline{K}/K)$ which is anticyclotomic, i.e. $\chi^{\sigma}:=\chi(\sigma g ...

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vote

**1**answer

155 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 ...

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926 views

### Solvability in differential Galois theory

It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative.
The proof I know goes as follows:
Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension ...

**6**

votes

**2**answers

515 views

### Polynomial with Galois Group $D_{2n}$

How does one construct a polynomial with Galois Group $D_{2n}$? A general method would be preferable or if that's impractical then an example of it being done for any n would be appreciated.
Thanks!

**15**

votes

**1**answer

866 views

### The Galois group of a random polynomial

Intuitively, the Galois group (of a splitting field over $\mathbb Q$ of) a polynomial $f\in\mathbb Q[X]$ taken at random is most probably the full permutation group on the roots of $f$. This intuition ...

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609 views

### Branch locus of the Galois closure of a Belyi morphism

A morphism of curves is said to be Galois if the corresponding extension of function fields is Galois.
Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective connected curves over ...

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195 views

### Involution on sextic polynomials?

The strangeness of $Aut(S_{6})$ suggests the following question:
Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ be the zeros of a 6th degree polynomial $P$ over a field $F$.
Let ...

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vote

**1**answer

205 views

### Image of spliting of short exact sequence of algebraic fundamental groups

If we have a variety X , over a field k, and x is a geometric point of X, and let $\bar x $be a geometric point of $X_{k^s} := X \times_k k^s $above x then we have the following short exact sequence:
...

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324 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 ...

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votes

**1**answer

359 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 ...

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votes

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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 - ...

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### 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 ...

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votes

**2**answers

418 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 ...

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votes

**1**answer

349 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 ...

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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 ...

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**2**answers

368 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: ...

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**0**answers

442 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 ...

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411 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 $L$
...

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votes

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290 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 $p$ ...

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876 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) ...

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348 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 ...

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235 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 ...

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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)
$n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ...

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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 $c\neq 0$.
...

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539 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 ...

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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 ...

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**1**answer

462 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 ...

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428 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
...

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### 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 ...

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votes

**1**answer

206 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 ...

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284 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 ...

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votes

**2**answers

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 ...

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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 ...

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### 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 ...

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793 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 ...

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### 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) ...

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votes

**1**answer

637 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 ...

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867 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?

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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 ...

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votes

**2**answers

817 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 ...

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2k 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 ...