Tagged Questions

4
votes
3answers
41 views

Automorphism of $K$ extending to $K[x_1,\dots,x_n]$.

I think it is well known that if $K$ is field, and $\sigma \in Aut(K)$, then the extended map $\sigma: K[x,y] \rightarrow K[x,y]$, given by $\sum a_{ij}x^iy^j \mapsto \sum \sigma(a_{ij})x^iy^j$ is ...
3
votes
1answer
63 views

Galois group of a quintic

What is the Galois group of $x^5-4x+12$? I'm able to show it has to be either the Frobenius $F_{20}$ group, or the dihedral group $D_{10}$. Is there a less computationally heavy way to determine? What ...
2
votes
1answer
40 views

Show the extension is not cyclic

Let $D\in\mathbb{Z}$ be a squarefree integer and let $a\in\mathbb{Q}$ be a nonzero rational number. Please show that $\mathbb{Q}(\sqrt{a\sqrt{D}})$ cannot be a cyclic extension of degree 4 over ...
5
votes
1answer
88 views

Unsolvability of $S_{n}$

Is there a short proof for unsolvability of $S_{n}$ without the standard approach of proving the simplicity of $A_{n}$ ? This is good, however, and one can prove this only with basic group theory, ...
0
votes
1answer
243 views

On Galois Theorem and Dummit and Foote text !

in dummit and foote text , galois theory is presented in chapter 14 . group theory is presented in 6 chapter , ring theory in 3 chapter and so on my qestion is , which chapters of the text is ...
4
votes
4answers
195 views

Necessarily complex analytic proofs in algebra.

Does anyone know of an example where complex analysis is necessary to prove something in algebra? I would be particularly interested in results from group theory or Galois theory. In an ideal ...
2
votes
1answer
94 views

Do we have such a direct product decomposition of Galois groups?

Let $L = \Bbb{Q}(\zeta_m)$ where we write $m = p^k n$ with $(p,n) = 1$. Let $p$ be a prime of $\Bbb{Z}$ and $P$ any prime of $\mathcal{O}_L$ lying over $p$. Notation: We write $I = I(P|p)$ to denote ...
6
votes
1answer
107 views

What does this notation mean: $\displaystyle\lim_{\leftarrow} \,\mathbb{Z}/n\mathbb{Z}$?

Just a small notation question from this Wikipedia page: The absolute Galois group of a finite field $K$ is isomorphic to the group $$\hat{\mathbb{Z}}=\lim_{\leftarrow} \mathbb{Z}/n\mathbb{Z}.$$ ...
3
votes
1answer
98 views

What can be said about the Galois group of $f(g(x))$?

Supposing we know the Galois groups of $f(x)$ and $g(x)$ over $K$, what can be said about the Galois group of $f(g(x))$? I suppose we can restrict the question to normal polynomials over ...
1
vote
0answers
76 views

Closure of Normal Subgroups of the Galois Group for an Infinite Galois Extension is Normal

Let K/F be an infinite Galois extension, and let N be a normal sub-group of Gal(K/F). Show that N closure is a normal subgroup of Gal(K/F).
17
votes
2answers
573 views

Is every group a Galois group?

It is well-known that any finite group is the Galois group of a Galois extension. This follows from Cayley's theorem (as can be seen in this answer). This (linked) answer led me to the following ...
3
votes
2answers
169 views

An explicit calculation of Galois group

This is a question requires to compute the Galois group of $X^4+1$ over $\mathbb{Q}$, $\mathbb{Q}(i)$, $\mathbb{F}_3$ and $\mathbb{F}_5$. Here is a brief of what I can think of. For the first two, ...
2
votes
2answers
172 views

If $N$ is a normal subgroup of $G$ with $N$ and $G/N$ solvable, prove that $G$ is solvable

I'm trying to prove that if $N$ is a normal subgroup of $G$, with $N$ and $G/N$ solvable, then $G$ is solvable. Proving that $G/N$ is abelian would of course suffice, but I'm not sure if that's a ...
4
votes
2answers
198 views

$p$-Sylow subgroups of a group of order $5^3\cdot 29^2$

I need to calculate the $p$-Sylow subgroups of a Galois group with order $5^3 \cdot 29^2$, i.e. $|\mathrm{Gal}(K/F)|=5^3 \cdot 29^2$. I've already established that there is only one 29-Sylow-subgroup ...
4
votes
1answer
216 views

Intermediate fields of cyclotomic splitting fields and the polynomials they split

Consider the splitting field $K$ over $\mathbb Q$ of the cyclotomic polynomial $f(x)=1+x+x^2 +x^3 +x^4 +x^5 +x^6$. Find the lattice of subfields of K and for each subfield $F$ find polynomial $g(x) ...
2
votes
1answer
262 views

Galois group of a degree 5 irreducible polynomial with two complex roots.

Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree five with exactly three real roots and let $K$ be the splitting field of $f$. Prove that Gal$(K/Q) \cong S_5$. My attempt: The ...
1
vote
1answer
202 views

Formal derivatives over finite fields.

I am slightly confused about what formal derivatives over finite fields mean. Example 1: Consider $f(x)=x^3-2\in \mathbb{F}_7[x]$. By checking each element of $\mathbb{F}_7$ we easily see that this ...
4
votes
1answer
189 views

Generating Elements of Galois Group

I am trying to prove the following: Let $K=\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n})$. Show that $K/\mathbb{Q}$ is Galois with Galois group $(\mathbb{Z}/2\mathbb{Z})^n$. I have attached my proof ...
1
vote
0answers
88 views

Abel-Ruffini Theorem Clarification

Let $n \geq 5$. I want to show that $\exists p\in \mathbb{Q}[X]$ of degree $n$ with roots which are not possible to find via radicals and rational functions from the coefficients of $p$. I've got the ...
2
votes
0answers
83 views

a computation in Galois theory

I am confused with some computation in Galois theory (this is not homework, just my weird curiosity). Let $k$ be a field of positive characteristic $p\neq 2$ that contains all roots of unity (e.g. ...
0
votes
1answer
160 views

How to generalize Dummit's resolvent for the quintic?

Dummit showed that given the five roots {$x_1, x_2, x_3, x_4, x_5$} of the quintic, then the expression, $x_1^2(x_2 x_5+x_3 x_4)+x_2^2(x_1x_3+x_4x_5)+x_3^2(x_1x_5+x_2x_4)+x_4^2(x_1x_2+x_3 ...
6
votes
0answers
423 views

Galois closure of a $p$-extension is also a $p$-extension

I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads: Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose Galois group is a ...
8
votes
3answers
630 views

Every finite abelian group is the Galois group of of some finite extension of the rationals

I'm trying to prove that every finite abelian group is the Galois group of of some finite extension of the rationals. I think I'm almost there. Given a finite abelian group $G$, I have constructed ...
3
votes
1answer
95 views

What's an example of an unsolvable finite group without an alternating group in its composition series?

The only examples I have seen have had an alternating group in the composition series.
3
votes
0answers
110 views

On Solvable quintics and septics

Here is a nice sufficient but not necessary condition on whether a quintic is solvable in radicals or not. Given, $\text{(1)}\;\;\; x^5+10cx^3+10dx^2+5ex+f = 0$ If there is an ordering of its ...
3
votes
1answer
145 views

surjectivity of group homomorphisms

I don't know if the next thing is true, but I'm not able to find a counterexample: suppose you have a surjective group homomorphism of finite groups $f:G \rightarrow G'$ and normal subgroups $H ...
3
votes
0answers
251 views

Galois group of solvable quintic is subgroup of Fr20

Why is it true that any solvable quintic polynomial in has a Galois group that is a subgroup on the Frobenius group of order 20? Thanks in advance.
1
vote
1answer
57 views

Subgroups of groups of order $2^{a-1}$

The context here is the following exercise Let $m=2^a$ with $a > 2$. Show that $\mathbb{Q}(\theta_m)$ contains exactly three quadratic subfields. By Galois theory, this reduces to the problem ...
2
votes
0answers
76 views

Is it true that $\forall G\leq Aut(\mathbb{P}^1)=PGL_2(\mathbb{C})$ the map $\mathbb{P}^1\rightarrow \mathbb{P}^1/G$ is defined over $\mathbb{Q}$?

Is it true that for every finite $G\leq Aut(\mathbb{P}^1_{\mathbb{C}})=PGL_2(\mathbb{C})$ the morphism $\mathbb{P}^1_{\mathbb{C}}\rightarrow \mathbb{P}^1_{\mathbb{C}}/G$ descends as a morphism (not ...
1
vote
1answer
62 views

$\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})$, where $m|n$

Find $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})$, where $m|n$. Is this proof correct? Since the base field is finite, $\mathbb{F}_{p^m}\subset \mathbb{F}_{p^n}$ is a Galois extension. (Is ...
7
votes
2answers
295 views

Solvable subgroups of $S_p$ of order divisible by $p$

This question is from Dummit and Foote's Abstract Algebra, page 638, question 20. It gives a nice paragraph of hints that basically guides one through the problem, but I'm very stuck at a crucial ...
23
votes
3answers
622 views

Galois groups of polynomials and explicit equations for the roots

Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
10
votes
3answers
506 views

A question regarding the definition of Galois group

In my book, Galois group is defined to mean the set of automorphisms on $E/F$ that "leave alone" the elements in $F$. On Wikipedia it says: "If $E/F$ is a Galois extension, then $Aut(E/F)$ is called ...
2
votes
1answer
149 views

Field extension question

A question I recall from Carl Linderholm's "Mathematics Made Difficult", Chapter 3 Exercise 8. ...
9
votes
4answers
448 views

How to describe the Galois group of the compositum of all quadratic extensions of Q?

This is Problem 1.7 from Gouvea's lecture notes on deformations of Galois representations. In particular, he asks you to show that it has many subgroups of finite index which are not closed. So here's ...