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 (galois-...

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I need help understanding this table for a Galois group

Here we are considering the field $E = \mathbb{Q}(\sqrt[3]{2}, \omega)$, where $\omega$ is a primitive cube-root of unity. The following table represents the Galois group $G(E/\mathbb{Q})$. I ...
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3answers
70 views

Finding a minimal polynomial of an algebraic element using Galois theory

There is a canonical (but difficult) way of determining the minimal polynomial of an algebraic element $\alpha$ in a field $F$, namely by considering the $F$-linear transformation defined by left ...
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1answer
59 views

Help justifying that $\mathbb Q(\sqrt[3]{2})$ is not a splitting field over $\mathbb Q$.

In a question related to introductory Galois theory, I was asked to given an example of a tower of fields $F \subset K \subset E$ such that $E$ is a splitting field for some polynomial $f(x) \in F[x]$,...
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0answers
36 views

A slight confusion in Galois Theory

Let $K=\mathbb{Q}$ and consider a cyclic extension $L$.(For example say, the splitting field of the polynomial say $f=x^3+x^2-2x-1$). Now consider a cyclotomic extension of $\Phi_3=X^2+X+1$. Let us ...
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2answers
53 views

$[k(\alpha):k]=p, [k(\beta):k]=q$, $p>q$ are primes, then $k(\alpha,\beta)=k(\alpha+\beta)$

Let $p>q$ be primes. Suppose $L\mid_{k}$ is an algebraic extension and $\alpha,\beta\in L$ are such that $[k(\alpha):k]=p$, $[k(\beta):k]=q$ and characteristic of $k$ is coprime with $p$. Show that ...
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1answer
28 views

Finite fields and generators of Galois group [closed]

I'm trying to find the order and all generators of the Galois group of $GF(256)$ over $GF(16)$? How to approach this problem?
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2answers
69 views

Galois extension of degree $2^n$ [closed]

Let $k$ be a field and $L=k(\sqrt{a_1},\sqrt{a_2},\cdots,\sqrt{a_n})$ with $a_i\in k$. Suppose $[L:k]=2^n$. Let $a=\sqrt{a_1}+\sqrt{a_2}+\cdots\sqrt{a_n}$. Show that $L=k(a)$.
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0answers
38 views

Group of polynomial $x^4+2$ in $\mathbb Q[x]$

Describe the Galois group of the polynomial $x^4+2 \in \mathbb Q[x]$. I've been able to see how to do this for $x^4-2$ and $x^4+1$ but am unsure how to do this for the polynomial above. Based on the ...
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1answer
31 views

How to prove $t(x)-x\in F_{p}$ when $p(x) \in K$, $K$ a field whose characteristic is $p>0$

Suppose $K$ is a field whose characteristic is $p>0$, and $L$ is a finite Galois extension of $K$. Also, we have an endormorphism $h$ such that $h(x)=x^p-x$ for any $x \in L$. Question: If $t\...
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1answer
26 views

Show that $[E:F] \le n!$ [duplicate]

Let $f(x)$ be a separable irreducible polynomial of degree n with coefficients in a field F. Let E be a splitting field of f(x) over F. Show that $[E:F] \le n!$
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When does a splitting field of a polynomial have the same degree as the polynomial?

Let's say we have the irreducible polynomial $f$ with roots $\alpha_1,\ldots,\alpha_n$. Now let $K$ be its splitting field, in other words $$K=\mathbb Q(\alpha_1,\ldots,\alpha_n).$$ When is it the ...
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1answer
44 views

The fixed field of $A$ is equal to the fixed field of $\langle A\rangle$.

Let $E$ be a finite extension field of $F$. Let $A$ be a subset of $Gal(E/F)$. Let $\langle A\rangle$ be the subgroup generated by $A$. Is the fixed field of $A$ equal to the fixed field of $\langle ...
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0answers
12 views

Galois group of the splitting field of a cyclotomic polynomial

Let $f(x) = x^4 + 1$,that is $f(x) = \Phi_8 (x)$. So one of the four primitive 8'th roots of unity is $w = e^{\pi i/4}$, giving that $E = \mathbb{Q}(w)$ is the splitting field of $\Phi_8(x)$. By a ...
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0answers
24 views

Extending the automorphism of $Q(\sqrt2)$ to automorphism of $Q(\sqrt(1+\sqrt2))$.

Question: does the automorphism $\alpha$ of $Q(\sqrt2)$ given by $\alpha(\sqrt2)=-\sqrt2$ extend to an automorphism of $Q(\sqrt(1+\sqrt2))$? In how many ways? My Answer is in the affirmative; i know ...
3
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1answer
68 views

Prove that there exists a normal extension $F/\mathbb{Q}$ with $G(F/\mathbb{Q}) \cong\mathbb{Z}_{5}$.

Prove that there exists a normal extension $F/\mathbb{Q}$ with $G(F/\mathbb{Q}) \cong\mathbb{Z}_{5}$. I tried to solve this problem by thinking about a polynomial which has a splitting field of ...
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0answers
37 views

Splitting field of a set of polynomials

Given $X\subseteq F[x]$ where $F$ is a field, how to prove that there exist a splitting field of $X$ over $F$? In the case that $X$ is finite, I think the answer can be solved using Kronecker's ...
2
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1answer
41 views

Is there a systematic way of finding the factorization over the closure of $\mathbb{Z}_2$ of $p(x) = x^{32} - x$?

And similar polynomials of the form $x^{p^n} - x$. I know that the degrees of the irreducible monic polynomials that factorize $x^{32} - x$ will have degree $d \vert 5 = 1, 5$. Also, I know that $x$ ...
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1answer
33 views

Find normal basis of the field $GF(3^6)$ and find the normal matrix

I am working with a homework is about normal basis on fields GF and I want opinions and maybe if you can help me in some doubts. 1) Find normal basis of the field $GF(3^6)$ which is understood as a ...
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1answer
26 views

Order of an orbit of Frobenius action on a algebraically closed field of characteristic p

Consider the action of the Frobenius homomorphism $F^{2}:\,\overline{\mathbb{F}_{q}}\rightarrow\overline{\mathbb{F}_{q}},\,x\rightarrow x^{q^{2}}$ over $\overline{\mathbb{F}_{q}}$ . Let $s=\left\{ ...
4
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0answers
30 views

The set of zeros of some polynomial over the closure of a finite field constitute a field in its own right

Let $p$ be a prime and $n$ be a positive integer and let $p(x) = x^{p^n} - x$ be a polynomial in $\mathbb{Z}_p[x]$. Let $Q$ be the set of all zeros of $p(x)$ over the algebraic closure of $\mathbb{Z}...
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1answer
46 views

Find the minimum polynomial of $u$ over $Q$ where $u=\sqrt3-(1+(5/2)^{1/3})^{1/4}$ [closed]

I tried using the binomial theorem but the terms keep increasing indefinitely
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1answer
63 views

Finding the splitting field of $x^3-5$ over $Z_7$ [closed]

My attempt; I substituted the values $0, 1, 2, 3, -1, -2, -3$ none of which yielded a zero, so I choosed (at random) $S_f=Z_{11}$ which gave only one zero
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0answers
23 views

What does "class $x \in X$ mean?

I'm working through Escofier's book on Galois Theory, and in several exercises regarding finite fields they use the terminology "class $x \in X$, and I'm not certain what it means. For instance, let $...
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1answer
41 views

Show that $\operatorname{Gal}(K/\mathbb Q)$ can be identified with the set of embeddings of $K$ into $\mathbb C$

I would be grateful if someone could help me demonstrate the following easy fact. Let $K$ be a number field which is Galois over $\mathbb Q$ and $\tau_0:K\hookrightarrow \mathbb C$ a fixed $\mathbb Q$...
2
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2answers
64 views

Galois group of splitting field

How can I compute the Galois group of the splitting field of the polynomial $x^4+x+1$ ?
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1answer
54 views

Isomorphic Galois groups imply isomorphic field extensions?

Suppose we have two field extensions $K/k$ and $L/k$. I am able to show that if these field extensions are isomorphic, then their corresponding Galois groups Aut$(K/k)$ and Aut$(L/k)$ are also ...
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0answers
26 views

Minimal polynomials over an inseparable extension

Let $F$ and $K$ be fields, and $\sigma$ and $\tau$ be two maps between them, $F \underset{\sigma}{\overset{\tau}{\rightrightarrows}} K$. Let $\alpha$ be an element algebraic over $F$, with minimal ...
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1answer
44 views

Why is the absolute value of this Gauss sum obvious?

I came across the Gauss sum discussed in the following post in a problem from my Galois theory course: http://mathoverflow.net/a/71282. Why exactly is the square of its norm obvious?
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3answers
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Show that the extension $F \vert \mathbb{Q}$ is Galois

Let $E \subset \mathbb{R}$ a field containing $\mathbb{Q}$ such that the extension $E \vert \mathbb{Q}$ is Galois, and let $F := E(\sqrt{-1})$. Show that the extension $F \vert \mathbb{Q}$ is Galois. ...
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0answers
61 views

Is my proof that $ \mathbb{Q}(\sqrt{5}) $ is the only quadratic subfield of $ \mathbb{Q}(\zeta_5) $ correct?

I would like some feedback on my proof of the fact stated in the question. First, we note that $ L = \mathbb{Q}(\zeta_5) $ is the splitting field of $ X^5 - 1 $, so it is Galois. Furthermore, we know ...
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2answers
56 views

Proving that a field is not a splitting field of any polynomial

I am trying to prove that $Q(2^{1/3})$ is not a splitting field of any polynomial $p(x) \in Q[x]$. I know I need to show that there does not exist any polynomial in $Q[x]$ that has all of its roots ...
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1answer
34 views

Showing automorphism group is trivial

Let $f(x) \in k[x]$ be a separable polynomial of degree $n\geq 3 $ with Galois group isomorphic to $S_n$, and let $\alpha \in \bar{k}$ be a root of $f(x)$. a) Show f is irreducible (I have already ...
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1answer
70 views

Find $\theta \neq \sqrt{3}+\sqrt{5}$ such that $\mathbb{Q}(\theta) = \mathbb{Q}(\sqrt{3},\sqrt{5})$. Need a hint to get started.

Here's what I've thought so far. Unless I'm very much mistaken, we have that $[\mathbb{Q}(\sqrt{3},\sqrt{5}):\mathbb{Q}] = 15$, so I'm looking for a $\theta$ such that $[\mathbb{Q}(\theta):\mathbb{Q}] ...
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2answers
30 views

Irreducible polynomials over $GF(4)$.

I'm working with $GF(4)$ and I'm looking for irreducible polynomials of different degrees over that field. So $GF(4) = \{0,1,\alpha, \alpha + 1\}$ where $\alpha^{2} + \alpha + 1 = 0$, and first I'm ...
2
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2answers
39 views

Galois group of a quartic which is also a quadratic in $x^2$

A few weeks ago a professor of mine mentioned that the Galois group of a certain type of quartic polynomial is easy to calculate, and at the time it seemed obvious to me so I didn't ask why. Now i'm ...
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1answer
39 views

how would you show that field automorphisms fix prime subfields?

Suppose K is a prime subfield of E, then if $\phi$ is an automorphism from E to E, we have for all x $\in$ K, $phi(x) = x$. I feel like this is just the definition of a field automorphism, but my ...
1
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1answer
38 views

Describing a Galois group for a field with the given roots adjoined

In this problem $\mathbb{Q}$ is viewed as a subfield of the complex numbers and a root $a = \sqrt[3]{3}$ is given. I find the min. polynomial for this root, which is $x^3 - 3$ and I factor it into: $(...
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1answer
14 views

Problem defining morphism in Galois cohomology of algebraic group

Let $K$ be a field and $G$ an algebraic group defined over $K$. If $M\supseteq L$ are two finite Galois extensions of $K$, then the groups $\text{Gal}(M/K)$ and $\text{Gal}(L/K)$ act, respectively, on ...
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1answer
45 views

Give an example of a polynomial whose Galois group is isomorphic to $D_{10}$

I have been spending my leisure time determining the subfield lattices and corresponding Galois subgroup lattices of some splitting fields of polynomials. I have made the lattice diagrams for the "...
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1answer
41 views

Reference needed with derivation of the equations for the cubic and quartic and proof of impossibility of quintic equation

My background is an undergrad year long course in algebra which got me through to some basic finite field and field extension stuff (most of fraliegh or Ian Stewart's book are familiar to me if that ...
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0answers
75 views

Galois theory for beginners, mistake?

I read this short article http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf. The goal of this article is to show the unsolvability of certain polynomials of degree $\geq 5$. But at the ...
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1answer
45 views

Trancendental extension Galois group

Let $K$ be a field and consider the extension $K(X)$ of rational functions with coefficients in $K$. It is common knowledge that $\text{Gal}(K(X)/K)$ is isomorphic to the group ${PL }_2(K)$, which is ...
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1answer
38 views

Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

An extension $K/F$ is Galois if and only if $K$ is the splitting field of some separable polynomial over $F$. Is there an example of some non-Galois extension $K/F$ where a (necessarily) non-separable ...
4
votes
1answer
43 views

Brauer group of cyclic extension of the rationals

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb Q^*/N(L^*...
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1answer
64 views

$K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Suppose $K$ is a finite extension of $\mathbb{Q}$. Suppose there is some complex number in $K$. Is it necessary that $\tau$, complex conjugation, is a member of $\text{Aut}(K/\mathbb{Q})$? I feel ...
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1answer
41 views

Degree of field extension over a fixed field

Suppose $L$ is a field and $H$ is a finite subgroup of $Aut(L)$. Then $[L:L^H]=|H|$ I've seen a proof of this that uses the Rank-Nullity theorem but it's rather cumbersome and difficult to remember ...
2
votes
1answer
70 views

Extensions of the quadratic closure of $\mathbb{Q}$

I'm looking for extensions of the quadratic closure of $\mathbb{Q}$ of degree $3$, degree $5$. Furthermore, Does there exist an extension of degree $4$? What I know: I know that the ...
0
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1answer
35 views

$x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is finite Galois extension with Gal$(F(x)/F)$ abelian of exponent $2$

Let $F$ be a field of characteristic that is not $2$. I want to prove that $x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is a finite Galois extension for which the group Gal$(F(x)/...
0
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0answers
93 views

Why is the Galois group of the polynomial $f(x)=x^5-x-1$ isomorphic to $S_5$?

Currently I am trying to understand why quintic (and higher degree) polynomials are not soluble by radicals, and one of the easiest examples of a quintic polynomial which has a nonsoluble Galois group,...
2
votes
2answers
42 views

Galois groups over p-adic fields

Let $k$ be a finite Galois extension of $\mathbb{Q}_p$. Do we know what groups $G$ show up as $G=Gal(k/\mathbb{Q}_p)$, as $p$ varies?