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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Intersection of two subfields of $F(X)$ [duplicate]

Let $E=F(x)$ for a field $F$ of characteristic $0$. Show that $F(x^2) \cap F(x^2-x) = F$ as subfields of $F(x)$. I could use a hand with this... Thanks
1
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1answer
27 views

Galois extension of intersection of fields

I have finite Galois extensions: E/K and E/L. $$M:=K \cap L$$ I am trying to prove that if the extension E/M is finite then it is also Galois. Any suggestions? Thanks
0
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1answer
29 views

The degree of the field extension $\mathbb{Q}(\sqrt{5},w): \mathbb{Q}$

Compute the degree of the field extension $$\mathbb{Q}(\sqrt{5},w): \mathbb{Q} ,$$ where $ w = e^{2\pi i / 3}$. I consider the tower of fields $\mathbb{Q} \subset \mathbb{Q}(\sqrt{5}) \subset ...
8
votes
1answer
105 views

Can $\cos (2\pi/7)$ be written as $p+\sqrt{q}+\sqrt[3]{r}, p,q,r\in \mathbb{Q}$?

Is it possible to find $p,q,r \in \mathbb{Q}$ such that $$\cos \frac{2\pi}{7}=p+\sqrt{q}+\sqrt[3]{r}.$$ Assume we can find such $p,q,r$, then $\mathbb{Q}(\cos \frac{2\pi}{7}) \subseteq ...
0
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1answer
22 views

Show that Gal$(E/\mathbb{Q})$ is abelian, where $E$ is the splitting field of $f(x)=x^{14} - 1$

Let $E$ denote the splitting field of $f(x)=x^{14}-1$. I want to show that the Galois group is abelian. Here's my attempt: The different 14'th roots of unity are given by $w=e^{i \pi n/7}$ where $n = ...
0
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1answer
17 views

Show that $\mathbb{Q}(\gamma)$ (where $\gamma$ is a primitive 5-th root of unity) is a splitting field for $x^5-1$

I know that my roots are going to be $\{1, \gamma, \gamma^2, \gamma^3, \gamma^4 \}$. I have to show that these are unique roots of the polynomial. In this problem I have to consider the polynomial ...
1
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1answer
47 views

Why is the Galois group for a cyclotomic extension isomorphic to $Z_n^{\times}$ and not to $Z_{n-1}$?

I'm struggling to understand this. Let $\mathbb{Q}(\xi)$ be the splitting field for $x^n - 1$, so here $\xi$ is the primitive n-th root of unity. If I consider the possible automorphisms of this ...
0
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0answers
23 views

Is $\prod_{\sigma\in \Sigma(L)}\mathbb Z$ free of rank $[K:\mathbb Q]$?

Let $L/\mathbb K$ be a Galois extension of number fields and $\Sigma(L)$ the set of embeddings of $L$ into $\mathbb C$. Let $H_L= \prod_{\sigma\in \Sigma(L)}\mathbb Z$. Let $G$ act on $H_L$ in the ...
1
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1answer
35 views

constructing a Galois group for a cyclotomic extension of $\mathbb{Q}$

Suppose that our polynomial is $x^5-1$, thus the splitting field is $\mathbb{Q}(\gamma)$ where $\gamma$ is a primitive 5-th foot of unity. Then our basis for the extension field will be: $\{1, ...
3
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1answer
69 views

Every finite group is isomorphic to the Galois group of some polynomial

I was reading through chapter 14 of Dummit and Foote just now and I came across the sentence "It is an open problem to determine whether every finite group appears as the Galois group for some ...
1
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1answer
26 views

Showing that a field $k$ is a splitting field for $p(x) \in \mathbb{Q}$

Suppose $\gamma$ is the fifth root root of unity. That is, $\gamma = e^{\frac{2\pi i}{5}}$, so $\gamma$ is a root of $p(x) = x^5-1$, or more precisely of $x^4+x^3+x^2+x+1$ since we can factor out a ...
2
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0answers
50 views

Embedding a number field in $\mathbb{Q}_p$.

Let $K/\mathbb{Q}$ be a finitely generated field extension and let $(x_1,\cdots,x_m)$ be a transcendance basis of $K/\mathbb{Q}$. Using primitive element theorem, there exists $y\in K$ algebraic over ...
0
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0answers
15 views

Calculating the fixed subfield of a splitting field $E$ corresponding to a subgroup of the Galois group $G = G(E/\mathbb{Q}$)

Here my splitting field is $E = \mathbb{Q}(\sqrt[3]{3}, \gamma)$, where $\gamma$ is a primitive cube-root of unity. This is the splitting field for $x^3-3$ in $\mathbb{Q}[x]$. I have calculated ...
4
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1answer
55 views

Abelian Galois group of $f$ implies splitting is simple extensions by a root of $f$.

Given an irreducible polynomial $f\in \mathbb{Q}[x]$ with Abelian Galois group, I would like to show that the splitting field $K$ over the rationals can be written as a simple extension ...
2
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0answers
72 views

How calculate Galois Group of $\mathbb{Q}_7(\zeta_3,\sqrt{3})/\mathbb{Q}_7$

From local field theory I know that if $L/K$ is a Galois extension of number fields, $\mathfrak{P}$ is a prime ideal of $L$ living above a prime $\mathfrak{p}$ of $K$, then the extension ...
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0answers
31 views

Is this a Galois extension?

I have the two simple extensions $F \subseteq F(\theta)$ and $F \subseteq F(\gamma)$, which are stated to be Galois extensions. We also have char$(F) = 0$. The problem is whether or not $F \subseteq ...
4
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1answer
43 views

Finding the Galois Group.

I am trying to find the Galois Group of $f(x)=x^4 + x^2 - 12$ over $\mathbb{Q}$. I was able to show that the factors $f(x)=(x^2-3)(x^2+4)$ were irreducible over $\mathbb{Q}$ and that the splitting ...
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2answers
34 views

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 ...
2
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3answers
63 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
55 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 ...
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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 ...
3
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2answers
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$[k(\alpha):k]=p, [k(\beta):k]=q$, $p>q$ 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
27 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
68 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
35 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
28 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 ...
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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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0answers
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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 ...
0
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1answer
42 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
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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 ...
0
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0answers
35 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
40 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 $x$ ...
0
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1answer
29 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 ...
1
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1answer
21 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
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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 ...
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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
57 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
22 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 ...
2
votes
2answers
60 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
50 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
23 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 ...
4
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0answers
55 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 ...
3
votes
2answers
52 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 ...
2
votes
1answer
33 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 ...
1
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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}] ...