1
vote
1answer
22 views

Prove that $K=k(\alpha)$

Prove: If K\k is a Galois extension and $\alpha \in K$ with $\sigma(\alpha)\neq \alpha$ for all $\sigma \in Gal(K,k)\backslash \lbrace id_K \rbrace$, than $K=k(\alpha)$.
3
votes
0answers
34 views

Proving that a certain maximal subfield has countably infinite index

Let $K$ be a maximal subfield of $\mathbb C$ which does not contain $\sqrt{2}$. I've shown that $\mathbb C$ is an algebraic extension of $K$, and that the Galois group of any finite extension of $K$ ...
2
votes
0answers
61 views

Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...
3
votes
0answers
29 views

Determining if a given equation is solvable given a set of ultra-radicals

So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots) AS WELL AS a set of inverses for some polynomials which are not solvable using ...
4
votes
1answer
37 views

Is the intersection of the conjugates of a subnormal subgroup of prime power index also a subgroup of prime power index?

I was wondering if it's really the case that, if $G$ is a group with subgroups $H$ and $N$ such that $H\unlhd N\unlhd G$ such that $G/N$ and $N/H$ is a $p$-group, then the intersection of all the ...
3
votes
2answers
130 views

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known?

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known? And if so is there a description on how the order can be found? My initial thoughts is that because $\bar{\Bbb Q}$ is countable and ...
2
votes
1answer
51 views

Characterisation of Galois Group with the action of $\sigma \in S_n$ on the roots

Let $f \in K[X]$ be irreducible and separable with roots $x_1,...,x_n$ in a splitting field $L$ of $f$ over $K$. We identify $\text{Gal}(L|K)$ with $\text{Gal}(L|K)\cong G\subset S_n$. How can I see ...
1
vote
1answer
46 views

Galois Group of a cubic polynomial

Let $P=X^3+a_1X^2+a_2X+a_3 \in \mathbb{Q}[X]$ be irreducible, $x_1,x_2,x_3$ the roots of $P$ and $L:=\mathbb{Q}[x_1,x_2,x_3]$. The galois group of $P$ is isomorph to $S_3$. Now we define ...
1
vote
0answers
28 views

How can i give this isomorphism between a Galois group and $S_3$?

I have shown that if $x$ is the nontrivial cubic root of the unity and that if $y$ is the real cubic root of $2$, then $Q(x,y)$ is a Galois extension whose Galois group has order $6$. I know that the ...
6
votes
1answer
69 views

Galois group of $X^5-n$

In the following situation I want to find the Galois group of a specific polynomial: Let $n > 1$ be a square-free integer, $f =X^5 - n \in \mathbb{Q}[X]$, $x:=\sqrt[5]{n}$ and $\zeta=\text{e}^{2 ...
1
vote
1answer
27 views

Automorphism group of a non-normal field extension

Consider finite field extensions $L>K>F$ such that $L/F$ is Galois, and $K/F$ is separable. I am particularly interested in the case $F=\mathbb{Q}$. By Galois theory, $K/F$ is normal iff ...
0
votes
1answer
44 views

Galois Group and Intermediate Field

I just need a detailed explanation of how to go about finding the intermediate fields and galois group of $x^4-x^2-6$. This is not a homework question, I am just confused on how to go about computing ...
1
vote
2answers
25 views

Difference between F-Automorphism and Identity morphism

In reading Shiffrin Abstract Algebra, in the section on Galois Theory, it gives the following definition: For K a field extension of F, a ring isomorphism $\phi:K\to K$ is an F-Automorphism if ...
1
vote
0answers
44 views

How to determine the Galois group of a general polynomial over rational number field?

How to determine the Galois group of a general polynomial over rational number field? For example $f(X)=X^n-X-3$, where $n$ is an positive number.${}$
6
votes
1answer
55 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
5
votes
1answer
72 views

Problem of Galois Extension

$\Bbb K$ is a non-Galois extension of $\Bbb Q$ and $[K:\Bbb Q]=4$. If $\Bbb F$ is the Galois closure of $\Bbb K$ then show that $Gal(\Bbb F/\Bbb Q)$ is either $S_4, A_4$ or $D_8$ with order 8. ...
3
votes
2answers
62 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
3
votes
1answer
66 views

Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
0
votes
1answer
42 views

A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
0
votes
1answer
53 views

Concerning a Cyclic Galois Group

Why is it that: $\forall K \supseteq \mathbb{Q}(\mathbb{i}), G=Gal(K / \mathbb{Q}) = \langle \sigma \rangle \implies \sigma (\mathbb{i}) = - \mathbb{i}$? (Note: I am guessing that $\sigma ≠ ...
1
vote
2answers
51 views

Constructing expressions

Suppose you are given all the elementary symmetric functions of $n$ variables $x_1,x_2,...,x_n$ and two rational functions $A(x_1,x_2,...,x_n)$ and $B(x_1,x_2,...,x_n)$ in the same $n$ variables that ...
2
votes
1answer
57 views

Galois group of intermediate fields

Find all subfields $M \subset \mathbb{Q}(\zeta_{7})$ where $\zeta_7=e^{2\pi i/7}$ and determine $Gal(\mathbb{Q}/M)$ and $Gal(M/\mathbb{Q})$. I've found that there are 2 intermediate fields ...
5
votes
1answer
50 views

The Four Transitive Subgroups of $A_7$.

I know that $A_7$ contains 3 transitive subgroups: $A_7, PSL(2,7), F_{21}$ (Alternating group of 7 elements, $PSL(2,7)$, Frobenius group of order 21). In studying the Galois group structure of ...
5
votes
2answers
66 views

Every finite Galois extension of $\mathbb{Q}_p$ has a solvable Galois group.

I have found the statement as in the title of this post on wikipedia, however, there is no reference for its proof. How does one prove it?
3
votes
2answers
495 views

Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
1
vote
2answers
56 views

Why $S_4$ has no transitive subgroups of order 6?

I know that every transitive subgroup of $S_4$ have to be order divisible by 4, but i should solve this with Galois Theory. I think this theorem can be usefull: Theorem 4.2. Let K be afield and f in ...
1
vote
0answers
45 views

Decomposition subgroup of Cyclic Galois Extension

My question: Say we have a cyclic Galois extension of degree $n$ over $\mathbb{Q}$. Denote the Galois group as $G$. If $H\leq G$, then does there exist a prime, $q$ in $\mathbb{Z} \subset \mathbb{Q}$ ...
5
votes
2answers
97 views

Elements of order 2 in the absolute Galois group

So I remember reading once that the only element of $G=Gal(\overline{\Bbb Q} / \Bbb Q)$ that we understand is complex conjugation. Suppose we fix an embedding of $\overline{\Bbb Q}$ into $\Bbb C$. ...
1
vote
1answer
90 views

prove that the Galois group $Gal(L:K)$ is cyclic

Let $L$ be a field extension of a field $K$. Suppose $L=K(a)$, and $a^n\in K$ for some integer $n$. If $L$ is Galois extension of $K$ (i.e., $L$ is the splitting field of $f(x)\in K[x]$, and $f$ is ...
5
votes
2answers
119 views

Infinite Galois Theory

In infinite Galois theory for the one-one correspondence as in the finite case, one needs to introduce Krull topology. What is the intuition behind defining such topology ?
2
votes
2answers
402 views

automorphisms of a finite field

Let $F$ be a finite field of characteristic $p$ over $\mathbb{Z}_p$ with $p^r$ elements. Then I have proved that the map $\phi: x\mapsto x^p$ is a field automorphism of $F$. Moreover, $\phi(x)=x$ if ...
1
vote
0answers
104 views

Galois Theory for Differential Equations?

Consider the set of algebraically primitive functions: that is the set of functions who can be created through some recursive expression involving arithmetic operators. Example $$y = x +1$$ $$y = ...
4
votes
1answer
143 views

Determine Galois groups of polynomials

Determine the Galois groups of the following polynomials over Q. $f(x)=x^3−3x+1$. $g(x)=x^4+3x+3$. $h(x)=x^5+8x+12$. I have no way to find their Galois groups. I only obtained that since they are ...
1
vote
3answers
339 views

Galois group of an irreducible polynomial

Find the Galois group of the polynomial $x^5-9x+3$ over $\mathbb{Q}$. since $3$ cannot divide $a_5$, $3$ divide other coefficients, $3^2$ cannot divide $a_0$, we see that the polynomial is ...
3
votes
1answer
131 views

Solvable by radicals or not?

Let $f=2x^5-5x^4+5\in \Bbb Q[x]$. Then, how to prove that it's not solvable by radicals? Since $f$ is solvable by radicals iff $Gal(f)$ is solvable and $Gal(f)\subset S_5$ (up to isomorphism), I ...
1
vote
0answers
46 views

Cyclotomic polynomials to find the subgroups of a Galois group

With $f(x) = x^{10}+1$, I want to draw the lattice of subgroups of the group $Gal(L/\mathbb{Q})$. Using cyclotomic polynomials I find that we have the $Gal(\mathbb{Q}(e^{\frac{2 \pi i}{20}}) / ...
1
vote
1answer
56 views

Galois groups of intermediate fields

Suppose $k\subset E$ is Galois and let $F$ and $F'$ be two intermediate fields. Let $FF'$ be the smallest intermediate field containing $F$ and $F'$. Also let $G$ denote $\text{Gal}(E/k)$. Let ...
1
vote
0answers
47 views

Determing the structure of the subgroup of an automorphism group

Suppose we have two automorphisms of an extension field $L=\mathbb{Q}(t)$ for some variable, given by $\sigma: t \mapsto 1-t$ and $\tau : t \mapsto \frac{1}{t}$. Clearly $\langle \sigma , \tau ...
4
votes
1answer
479 views

Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$

So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group. My thoughts are as follows: Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is ...
2
votes
0answers
73 views

Proving Lagrange's Theorem with Galois Theory

Problem: Let $G$ be a finite group with subgroup $H$. Let $|G| = n$ and $|H| = m$. Prove that the order of $H$ divides the order of $G$ using only results from Field and Galois Theory, with the ...
3
votes
1answer
124 views

Finite abelian unramified $p$-extension of a number field

Let $K$ be a number field. How many finite abelian unramified $p$-extensions of $K$ are there and what are their Galois groups? My feeling is, that every group $\mathbb{Z} / p^n \mathbb{Z}$ can occur ...
3
votes
0answers
137 views

Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly ...
3
votes
1answer
70 views

Show that $\mathbb{Q}(\zeta)$ is the Splitting Field for $x^n - 1 \in R_n[x]$

Let $R_n = \{\bar{x}$ modulo $n : (x,n) = 1\}$ which forms a group under multiplication. Let $p(x) = x^n - 1 \in \mathbb{Q}_n[x]$ have roots $\zeta_1, \zeta_2, \ldots , \zeta_n$. Prove that there is ...
2
votes
1answer
80 views

$[E:F]$ can be divided by $|Gal(E/F)| $?

Assume that $E$ and $F$ are fields, and $E$ is a finite extension over $F$. Then how to prove that $[E:F]$ can be divided by $|Gal(E/F)| $? I want to prove it by induction on $[E:F]$. For the case ...
5
votes
0answers
87 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
2
votes
0answers
39 views

A certain inverse limit

Let $p$ be an odd prime and $n$ a positive integer. Let $\zeta_{p^{n+1}}$ be a primitive $p^{n+1}$-th root of unity. It can be shown that $Gal(\mathbb Q(\zeta_{p^{n+1}})/\mathbb Q)\cong (\mathbb ...
2
votes
1answer
85 views

Splitting field of resolvent equals that of $f$

Lemma: Let $\Psi \in k[X_1,...,X_n]=:B$ be s.t. $stab_{S_n}(\Psi)=H \subset S_n$, $S_n/H=\{ \Psi, t_2 \Psi,..., t_e\Psi \}$, $\Delta_\Psi$ the discriminant of $L_\Psi:=\prod_{i=1}^e (X-t_i \Psi)$, and ...
4
votes
1answer
101 views

Possibilities for $[KL:F]$ when $[K:F]=[L:F]$ is prime

Suppose $K/F$ and $L/F$ are extensions of $F$ (contained in some common field) of degree $p$, where $p$ is prime. Standard arguments show that $[KL:F]$ must be in $\{p,2p,\ldots,p^2\}$. But are all ...
2
votes
0answers
101 views

Degree of factor in a resolvent

Background and lemma first: Let $\Theta \in k[x_1,...,x_n]^H$ and $\theta$ be its evaluation to the roots of a fixed $n$:th degree polynomial in $k[x]$. Put $L(t) = \prod_{\sigma \in S_n//H} (t-\sigma ...
2
votes
1answer
50 views

What could be said about a field extension, if it's Galois Group is fixpointfree?

Let $L/K$ be a galois extension, and let the Galoisgroup be fixpoint-free, what could be said about the field extension.