Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Uniformly solvable families of polynomials

It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
0
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3answers
58 views

automorphisms and field extension $E$ of $\mathbb{Q}$.

I want a hint. That is all I ask for. The question I am asked to prove is as follows: Let $E$ be an extension field of $\mathbb{Q}$. Show that any automorphisn of $E$ acts as the identity on ...
2
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0answers
192 views

Non-isomorphic simple extensions of the same degree of a field of positive characteristic

Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic. I thought of an example where they are ...
1
vote
1answer
394 views

Test for polynomial reducibility with binary coefficients

I'm learning about Galois Fields, in particular $GF(2^8)$, as they are applied to things like the AES algorithm and Reed-Solomon codes. Each of these rely on an irreducible 8th degree polynomial with ...
4
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2answers
71 views

Find a monic polynomial over $\mathbb{Q}$ whose Galois group is $V_4$

I need to find a polynomial over $\mathbb{Q}$ whose Galois group is $V_4=\langle(12)(34),(13)(24)\rangle\subset S_4$. I can find examples of such polynomials, but I wonder if one can construct such a ...
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2answers
71 views

GgT, (polynomial) division and finite fields…

Exercise: Let $f,g \in \mathbb{Z}_2[x]$ be the polynomials $f = x^6 + x^5 + x^4 + 1$ and $g = x^5 + x^4 + x^3 + 1$. Has the diophantic equation $f u + g u = x^4 + 1$ solutions $u,v \in \mathbb{Z}[x]$? ...
5
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2answers
113 views

Galois Extensions and $n^{\text{th}}$ Roots

I've been studying for my prelims lately, and this problem has me stuck: (a) Let $K$ be a field with no abelian Galois extensions. Suppose that $n$ is a positive integer and either $char(K)=0$ or ...
5
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1answer
123 views

finding fixed field of automorphism

Let $F$ be a field and let $g:F(x) \to F(x)$ be the automorphism which maps $x$ to $x+1$. I need to find the fixed field of this automorphism. So far I know $g$ fixes $F$. I want to use Galois ...
4
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2answers
72 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 ...
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0answers
95 views

Show that an field extension is algebraic (normal).

Let $A/K$ be a field extension, I wanted to proof: $A/K$ is normal iff for every irreducible polynomial $P \in K[x]$ which has a root in $A$, the field extension $A$ contains a splitting field for ...
4
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2answers
82 views

Solving equations by radicals

Let $\zeta$ be a complex number, $\zeta\neq 1, \zeta^3=1$. Then the expression $$(x_1+\zeta x_2+\zeta^2x_3)^3$$ takes only two distinct values when we permute $x_j$'s. $\bf{Why\ this?}$ Hence it ...
2
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1answer
480 views

Galois group of $x^8+2$ over $\Bbb{Q}$

This is what I did to find the Galois group for $x^8+2$: Splitting field: $$K = \Bbb{Q}(\zeta_8, \zeta_{16}2^{1/8})$$ Since $Gal(\Bbb{Q}(\zeta_8)/\Bbb{Q}) \cong Aut( \langle\zeta_8\rangle) \cong ...
2
votes
1answer
282 views

Calculating The Galois Group of the Splitting Field of $f=x^3-3$

If we let $f=x^3-3$ the let L be the splitting field for this polynomial I am trying to find $\Gamma(L:\mathbb{Q})$ and all intermediate field extensions. Now as this is a splitting field and finite ...
2
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1answer
101 views

Understanding how the Galois group acts on the character of a group representation

I'm trying to understand a proof sketch found here: http://mathoverflow.net/questions/10635/why-are-the-characters-of-the-symmetric-group-integer-valued If $g$ is an element of order $m$ in a ...
10
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1answer
250 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
2
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0answers
195 views

Is simple extension $\mathbb{Q}(\cos\pi/9):\mathbb{Q}$ algebraic and normal?

I am trying to prove this question: Q: show that the simple extension $\mathbb{Q}(\cos\pi/9):\mathbb{Q}$ is an algebraic extension and also normal extension? Thank you
4
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1answer
85 views

A Galois Theory Question

Fix a prime $p$ and consider the equation $X^p-X-t^{-1}$ over $\mathbb F_p((t))$, the field of formal Laurent series over $\mathbb F_p$. What is the Galois group of this equation? After fumbling ...
6
votes
1answer
626 views

Finding the Galois group over $\Bbb{Q}$.

If K is the splitting field of $X^8-2$ over $\Bbb{Q}$, I want to find the galois group. We know that $K=\Bbb{Q}(2^{1/8}, \zeta_8)$. So first I want to look at $Gal(\Bbb{Q}(\zeta_8)/\Bbb{Q})$ and ...
2
votes
1answer
256 views

Galois Group of $f(x)=x^4 - 10 x^2 + 1$

I am trying to calculate the Galois group of $f(x)=x^4 - 10 x^2 + 1\in\mathbb Q[x]$ over $\mathbb Q$. In my notes it says that the four roots are $\pm\sqrt 2\pm\sqrt 3$. So the splitting field of ...
3
votes
2answers
116 views

Show $\zeta_p \notin \mathbb{Q}(\zeta_p + \zeta_p^{-1})$

I asked a question here: [Writing a fixed field as a simple extension of $\mathbb{Q}$ ], but realised I couldn't justify why the given quadratic was irreducible. Thus: Is there a way of showing ...
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1answer
65 views

Writing a fixed field as a simple extension of $\mathbb{Q}$

I have $G = Gal(\mathbb{Q}(w) : \mathbb{Q})$ where $w = e^{2\pi i/p}$ for $p$ prime. I have that there exists a (unique) element in $G$ of order 2, say $\phi$. I'm trying to express the fixed field ...
3
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1answer
254 views

The Fundamental Theorem of Galois Theory.

Let E/F be a finite Galois extension with Galois group G. If H is a subgroup of G, let F(H) be the fixed field of H,and if K is an intermediate field,let G(K) be Gal(E/K), the fixing group of ...
2
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1answer
75 views

question on Galois theory

Can anybody help me with the following question ? I start with a number field $F/\mathbb{Q}$ which is abelian (that is, a Galois extension of abelian Galois group). I know by the Kronecker-Weber ...
6
votes
2answers
618 views

If $f(x)$ is an irreducible polynomial of degree n, then the cardinality of its Galois group is divisible by $n$.

If $f(x)$ is an irreducible polynomial of degree $n$, then the cardinality of its Galois group is divisible by $n$. I know I need to use the Tower Theorem, but I can't figure out how to get from ...
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1answer
304 views

Compute the Automorphisms of tower of fields

Suppose we have the following tower of fields: $\mathbb Q \subset \mathbb Q(\sqrt{2}) \subset \mathbb Q(\sqrt[4]{2})$. Compute Aut$(\mathbb Q(\sqrt{2})/\mathbb Q)$, Aut$(\mathbb Q(\sqrt[4]{2})/\mathbb ...
2
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3answers
53 views

Is $[L : K] = 2$, $f \in K[x]$ irreducible, then $\operatorname{deg}(f) \le 2$ valid?

Is it true that in every field extension of degree two, every irreducible polynomials has a degree smaller than two? And if so, how can I proof this?
0
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2answers
47 views

If $[L : K ] = n$, then for every irreducible polynomial $f$ is $\operatorname{deg}(f) \le n$.

If a field extension is finite with degree $n$, how can I proof that every irreducible polynomial has a degree smaller then $n$. I guess that this is valid, but I am not sure how to proof this?
2
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1answer
135 views

A basic question on factorization

Is the following true? If not, can anyone add some reasonable assumptions to make it true? Statement. Let $E/F$ be a field extension and let $\alpha \in E \setminus F$ be algebraic over $F$. If $f(x) ...
2
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2answers
923 views

Galois Group of an irreducible cubic

I need to prove that if the Galois group of an irreducible cubic over $\mathbb Q$ is $\{id, \sigma, \sigma^2\}$, then all the roots of the cubic are real. How do I even start this?
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0answers
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A question regarding linear disjiontness and the degree of a field extension

Let $K / k$ and $L / k$ be field extensions with $K$ and $L$ linearly disjoint over $k$. Suppose that $K' / K$ is an extension of finite degree, and that all fields are characteristic zero. Then is ...
5
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2answers
205 views

existence of an automorphism of $k^a$ whose fixed field is $k$

Let $k$ be a field such that every finite extension is cyclic. Show that there is an automorphism of $k^a$ over $k$ whose fixed field is $k$. Here $k^a$ is the algebraic closure of $k$. P.S. It's a ...
7
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1answer
146 views

Calculate a galois group

I am trying to calculate the galois group $\operatorname{Gal}( \mathbb{Z}_q (\vartheta_p) : \mathbb{Z}_q) $, where $p$ and $q$ are different primes, $\mathbb{Z}_q$ $q$-adic ring, $\vartheta_p$ a ...
1
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1answer
151 views

There could no multiplication on a three dimensional vector space defined similar to complex multiplication, but what an field extension of degree 3?

According to Wikipedia there could be no multiplication on a three dimensional vector space like the multiplication over the complex numbers. But what about field extensions of degree three, in these ...
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1answer
165 views

A question regarding the finiteness of the degree of a field extension

Let $x$ and $y$ be transcendental and $y$ be algebraic over $\mathbb{Q}(x)$. Let $F$ be an algebraic Galois extension of $\mathbb{Q}(x)$ of infinite degree and let $\mathbb{Q}^{al}$ be the algebraic ...
3
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2answers
116 views

If all embeddings of $E/F$ are automorphisms, then $E/F$ is normal

Let $E$ be a (possibly infinite) field extension of $F$, and let $\Omega$ be an algebraic closure of $E$. I'm trying to prove that if $\sigma(E)=E$ for all $F$-algebra embeddings $\sigma \colon E\to ...
11
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0answers
73 views

Field-theoretic description of fixed field of central subgroups?

Given a Galois extension $E/F$ with Galois group $G$, and a subextension $E/K$ with Galois group $H$, is there a "field-theoretic" characterization of when $H$ is central (i.e. $H\leq Z(G)$)? By ...
4
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2answers
167 views

3 questions on field extensions

I am trying to figure out some things regarding field extensions and some questions have arisen on the way. Let $a$ be a positive integer which doesn't have a rational $nth$ root: Is the splitting ...
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3answers
2k views

Quadratic subfield of cyclotomic field

Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
3
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1answer
106 views

Why don't I end up with the same splitting field?

I've understood that the splitting field of $x^4+2$ and the splitting field of $x^4-2$ over $\mathbb{Q}$ are both the field $\mathbb{Q}(\sqrt[4]{2} , i)$. With degree $8$ over $\mathbb{Q}$. This ...
2
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1answer
156 views

Splitting field for $x^n+a$

What is a splitting field $E$ for $f(x)=x^n+a$ over the field $K$ of characteristic zero? If I put $g(x)=x^{2n}-a^2=(x^n-a)(x^n+a)$. The splitting field $F$ of $g(x)$ is $K(\sqrt[n]{a},\alpha )$ ...
3
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1answer
205 views

Galois group of irreducible quartic with real coefficients

Let $K$ be a subfield of the real numbers and $f\in K[x]$ be an irreducible quartic. If $f$ has exactly two real roots, show that the Galois group of $f$ is $S_{4}$ or $D_{4}$ (I'm using the ...
9
votes
1answer
351 views

Galois group of $X^5 - X^3 - 2X^2 - 2X - 1$ over $\mathbb{Q}$.

So far I have found that this polynomial is irreducible, and its discriminant is a square. Therefore the Galois group it a transitive subgroup of $A_5$. I found out that the only transitive ...
3
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1answer
148 views

Irreducible polynomial over $\mathbb Z/7\mathbb Z$

Is the polynomial $t^6-3$ irreducible over $\mathbb{Z}/7\mathbb{Z}$? How would you prove that without doing all the computations? Is there a general method for this? Thank you.
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2answers
91 views

Example where $[E:\mathbb{Q}]<|\mathrm{Aut}_{\mathbb{Q}}E|$

Let $F$ be an algebraic closure of $\mathbb{Q}$ and let $E\subset F$ be a splitting field over $\mathbb{Q}$ of the set $\{x^{2}+a|a\in\mathbb{Q}\}$ so that $E$ is algebraic and Galois over ...
2
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0answers
42 views

Prove (or disprove) that the Galois group of $X^4 - X^3 - 7X + 19$ is not $S_4$.

I have checked that $X^4 - X^3 - 7X + 19$ is coprime to its derivative. So it is separable. So I know that the Galois group of $X^4 - X^3 - 7X + 19$ must be a subgroup of $S_4$. But I need to prove ...
0
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1answer
70 views

Smallest Galois extension

Let $F$ be a finite dimensional Galois extension of $K$ and let $E$ be an intermediate field. Show that there is a unique smallest field $L$ such that $E\subset L\subset F$ and $L$ is Galois over $K$. ...
2
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1answer
161 views

Showing that a field extension is Galois

Let $F$ be an extension field of a field $K$. Let $E$ be an intermediate field such that $E$ is Galois over $K$, $F$ is Galois over $E$, and every $\sigma\in\mathrm{Aut}_{K}E$ is extendible to $F$. ...
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0answers
58 views

Is there a way to prove that $x^n+x^m=C, x$ $\&$ $C \in ℝ, n$ $\&$ $m \in ℕ$ has no general solution for $x$?

I think I have made myself clear in the title and is hoping for someone to show this with as simple math as possible. Since I'm only 17 and haven't learned all of the complex mathematics yet. However ...
3
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1answer
356 views

Lagrange interpolation of Galois field functions

I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
3
votes
1answer
129 views

A generalization of Galois connections

Let $f(x)\ast y \Leftrightarrow x\ast g(y)$ for some binary relation $\ast$ and functions $f$ and $g$. This is a generalization of Galois connections. Are things like this studied before? Note that ...