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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Splitting field of cyclotomic polynomials over $\mathbb{F}_2$.

Let $\Phi_5$ be the 5th cyclotomic polynomial and $\Phi_7$ the 7th. These polynomials are defined like this: $$ \Phi_n(X) = \prod_{\zeta\in\mathbb{C}^\ast:\ \text{order}(\zeta)=n} ...
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1answer
33 views

Char(F)=0, K/F is abelian and F contains a primitive root of unity

$\mathrm{Char}(F)=0, K/F$ is abelian . Let $n$ be a positive integer such that $f^n=1$ for every $f$ in $G(K/F)$ and $F$ contains a primitive $n^{th}$ root of unity. Prove there exist $x_1, x_2, …, ...
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votes
2answers
77 views

Subgroup of class group

Let $K/\mathbb Q$ be a finite Galois extension with Galois group $G$, ring of integers $\mathcal O_K$ and $\mathcal Cl(K)$ its ideals class group. I want to show that $\mathcal Cl(K)^G$ is generated ...
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0answers
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M/F normal, determine whether there is an automorphism $\sigma\in Gal(M/F)$ with $\sigma(a)=a'$ and $\sigma(b)=b'$

Let $M$ be a normal extension of $F$. Suppose that $a, a'$ are roots of $min(F,a)$ and that $b,b'$ are roots of $min(F,b)$,and that $min(F,a)\neq min(F,b)$. Determine whether or not there is an ...
31
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1answer
602 views

Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree. Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$? ...
3
votes
1answer
48 views

Automorphisms of a field extension permute roots of irreducible factors.

Would someone mind confirming (or refuting) the following... Proposition. Let $K$ be a field and let $f \in K[X]$. Suppose $g \in K[X]$ is an irreducible factor of $f$. Let $L$ be a splitting field ...
2
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1answer
79 views

$L=K(\alpha)$ with $\alpha^p-\alpha=a\in K$

Let $K$ be a field of characteristic $p$. $L=K(\alpha)$ with $\alpha^p-\alpha=a\in K$, an extension of order $p$. Show that there does not exist $\beta \in L$ such that $\alpha^{p-1}=\beta ^p-\beta$. ...
2
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1answer
74 views

Having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space over $\mathbb{F}_p$.

As the title states, I'm having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space V over $\mathbb{F}_p$. Clearly it is dimension $n$. How can we work with this vector space? ...
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0answers
25 views

E is a Galois extension of F

15.32 Let $E$ be a finite extension of a field $F$ with dimension $n$. Show that $|\operatorname{Gal}_F(E)| = n$ if, and only if, $E$ is a Galois extension of $F$. Please some help to find the ...
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0answers
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Field extension, primitive root of unity solvable by radicals

I'm stuck with this exercise. I'd like to show, that for a field $K$ with $\mathrm{char}(K)=0$ and $\omega_n$ being a primitive n-th root of unity $K(\omega_n)/K$ is solvable. I thought I could do ...
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1answer
27 views

Galois group of $K(\sqrt[p]a)$ over $K$

If $p$ is a prime number and $a$ is an element of the field $K$, what can be said about the Galois group of $K(\sqrt[p]a)$ over $K$? (That is, $\sqrt[p]a$ is an element whose $p$-th power is $a$) I ...
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votes
1answer
50 views

A question about Galois extension

Let $n\geq 2$. Let $K$ be a field which contains $n$ distinct n-th roots of unity. Let $L=K(\sqrt[n]{a})$ with $a\in K$ and $[L:K]=n$. Show: there exists a Galois extension $K\hookrightarrow M$ such ...
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0answers
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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
71 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 ...
3
votes
1answer
31 views

Minimal polynomial in Galois extension

Let $K \subset L$ be a Galois extension and $x \in L$ such that $L=K[x]$. $H \leq \text{Aut}(L|K)$ is a subgroup of the galois-group. I want to show that the minimal polynomial of $x$ over ...
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0answers
131 views

If $f(x)\in F[x]$ is solvable by radicals then its Galois group $Gal(E/F)$ is solvable.

In all the books that I've checked the use of roots of unity (as hypothesis) is very crucial to prove that if $f(x)\in F[x]$ is solvable by radicals then its Galois group $Gal(E/F)$ is solvable, but ...
1
vote
1answer
23 views

A Galois extension of two roots.

Let $x$ be one of the nontrivial cubic roots of the unity and let $y$ be the cubic root of $2$. Prove that $\mathbb{Q}(x,y)$ is a Galois extension and then, show that is isomorphic to one of the ...
5
votes
2answers
93 views

Galois group of irreducible Quartic polynomial over $\mathbb{Q}$

Actual Question is : What are all possible galois groups of an irreducible Quartic polynomial over $\mathbb{Q}$ As polynomial is irreducible, Galois group is transitive subgroup of $S_4$. I ...
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1answer
39 views

Problem with a Galois extension [duplicate]

How can i prove that Q(a,b) is a Galois extension and that its Galois group is of order 6, if a is a root of the cubic that is not 1 and b is the cubic root of 2? Can you give me a little hint for ...
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3answers
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$\mathbb{Q}(\sqrt[3]{2}, \zeta_{9})$ Galois group

How do I calculate the degree of $\mathbb{Q}(\sqrt[3]{2}, \zeta_{9})$ over $\mathbb{Q}$. Should it be 18, as $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}] = 3$, and $[\mathbb{Q}(\zeta_{9}):\mathbb{Q}] = 6$? ...
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vote
1answer
36 views

Primitive Element and Minimal Polynomial

Let $K = \mathbb{Q}(\sqrt{2},\sqrt{5},\sqrt{7})$: I know that $K/F$, with $F =\mathbb{Q}$, is Galois and has degree 8. By the primitive element theorem there is a a $\theta$ such that $K = F(\theta)$, ...
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1answer
14 views

There are finite distinct restrictions to a subfield

Consider the field extension $L\subseteq K\subseteq \mathbb C$ where $K/L$ is finite. I must show that the set $\{\sigma_{|K}\,:\,\sigma\in\operatorname{Gal}( \mathbb C/L)\}$ is finite, but I have ...
5
votes
1answer
190 views

Galois extension with galois group $A_4$

Suppose $char F \neq 2$ and $K/F$ is a degree three galois extension with $Gal(K/F)\cong \mathbb{Z}/(3)$. Is there a bijection between extensions $N/F$ with galois group $A_4$ and the order four ...
5
votes
1answer
92 views

Galois Group of $x^{14}+x^7-1$ over $\mathbb{Q}$

So consider the polynomial $f(x)=x^{14}+x^7-1$ defined over $\mathbb{Q}$. We want to determine its Galois Group. So let's look for the splitting field, $L$ say, to give us an idea of the size of the ...
3
votes
1answer
53 views

Galois Group of $x^4+x-1$ over $\mathbb{F}_3$

Consider the finite field $\mathbb{F}_3$ and define the polynomial $f(x)=x^4+x-1$ over $\mathbb{F}_3$. I want to find its Galois Group. I observe that $f$ has no root over $\mathbb{F}_3$, so if it ...
1
vote
1answer
30 views

maximal abelian extension of exponent $q-1$ of $\mathbb F_q((t))$

I would like to find the maximal abelian extension of exponent $q-1$ of $K=\mathbb F_q((t))$ and find its Galois group. Due to Kummer theory this extension is $K(\sqrt[q-1]{K^*})$ and it's Galois ...
30
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1answer
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Is Frobenius the only magical automorphism?

The Frobenius automorphism is special because the $p$-power map makes sense in any characteristic $p$ ring, which allows us to canonically extend the Galois-theoretic Frobenius to any such ring. I ...
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K/F is finite separable extension show that N_K/F = N_E/F * N_K/E and Tr_K/F = Tr_E/F * Tr_K/E for any intermediate field K/E/F.

K/F is finite separable extension show that $N_{K/F} = N_{E/F} N_{K/E}$ and $Tr_{K/F}$ = $Tr_{E/F}Tr_{K/E}$ for any intermediate field K/E/F If x is an element of K then $N_{K/F}$ is the norm ...
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0answers
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MAGMA Commands for Galois Theory calculations

Perhaps this is walking over old ground (or the wrong place to ask this), but I'm looking to use MAGMA to perform certain calculations in Galois Thoery. The motivation of this question is to create an ...
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0answers
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Prove if $F$ is an infinite field and $u,v$ are algebraic and separable over $F$ that there exists $x \in F$ such that $F(u,v) = F(u+xv)$.

Prove if $F$ is an infinite field and $u,v$ are algebraic and separable over $F$ that there exists $x \in F$ such that $F(u,v) = F(u+xv)$. Can someone give any hint? Thanks
3
votes
3answers
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Show [F(a):F] = r

Suppose that K/F is Galois and a in K has precisely r distinct images under G(K/F). Show [F(a):F] = r I'll really appreciate if someone could give some hints? Thank you!
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2answers
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When has a polynomial a non-real and a real root?

Let $g \in \mathbb{Q}[X]$ be an irreducible and separable polynomial which has a real and a complex root in $\mathbb{C}$. Show that in this situation $\text{Gal}(K|\mathbb{Q})$ is not abelian, where ...
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1answer
28 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 ...
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0answers
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Why G(P/F) and G(Q/F) abelian implies PQ abelian?

Suppose L/F is a field extension, then L/P/F with P/F abelian and L/Q/F with Q/F abelian implies PQ=P(Q) abelian. After trying to pick element from PQ, I still can't find the relation between PQ and ...
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votes
4answers
211 views

Learning about Grothendieck's Galois Theory.

I have background in category theory and I am familiar with the very basics of algebraic geometry - Chapters I and II of Hartshorne. What would be a recommended (self-contained, maybe?) text for ...
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Question: Let L/F be a field extension and L/P/F with P/F abelian and L/Q/F with Q/F abelian then PQ=P(Q) .

Question: Let L/F be a field extension and L/P/F with P/F abelian and L/Q/F with Q/F abelian then PQ=P(Q) is abelian. After trying to pick element from PQ, I still can't find the relation between PQ ...
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2answers
57 views

Construct an irreducible polynomial of degree $3$ over $\mathbb{Q}$

I'm trying to construct an irreducible polynomial of degree 3 with rational coefficients such that this irreducible polynomial has one root in the real numbers and has two complex roots. Everything I ...
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1answer
21 views

irreducible polynomial is still irreducible on purely inseparable extension

I wish to prove that $\alpha \in C$ where $C$ is algebraically closed field of $F$, $f(t) \in F[t]$ is minimal monic irreducible, separable polynomial having $\alpha$ as a root. Then let ...
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0answers
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Reference for Galois Theory of infinite field extensions.

I would like to ask what is in your opinion the best place to learn about Infinite Galois Theory that requires not much knowledge of topology. I am searching for a text that explains the notions ...
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1answer
25 views

A group of order less than $60$

I'm given a theorem (call it Theorem 1) which states that $G$ is solvable iff there exists a chain of normal subgroups $\{e\} = G_0 \unlhd \ldots \unlhd G_n = G$ such that $G_i / G_{i-1}$ is abelian ...
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0answers
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Consider $n$ a product of distinct primes $p_j$, with each $p_j-1$ a power of 2, then the regular $n$-gon is constructible.

I am having trouble showing that if $n$ is a product of distinct primes $p_j$, with each $p_j-1$ a power of $2$, then the regular $n$-gon is constructible. Apparently it could be useful to use that ...
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vote
3answers
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Find a group isomorphic to the Galois group of the polynomial $x^3+4$?

Find a group isomorphic to the Galois group of the polynomial $x^2+4$? Am I correct that it will be isomorphic to $S_3$?
3
votes
2answers
156 views

Characteristic Polynomial of Galois automorphism

Let $K/F$ be a finite Galois extension. Let $g$ be an element of $Gal(K/F)$ How do I compute the characteristic polynomial of $g$, where $g$ is considered as a $F$-linear map $K \rightarrow K$?
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votes
1answer
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Galois Extension of Even Order

Suppose $E=F(\alpha)$ is a proper Galois extension. Let $\sigma \in Gal(E/F)$ such that $\sigma(\alpha)=\alpha^{-1}$. Show that $[E:F]$ is even and $[F(\alpha + \alpha^{-1}):F]=\frac{1}{2} [E:F]$. I ...
3
votes
1answer
72 views

Galois extension and morphism of curves

Let $\phi: C \rightarrow \mathbb P^1$ a morphism (over a field of characteristic 0) from a rational curve $C$ to $\mathbb P^1$ of degree 3. By the Riemann-Hurwitz formula the degree of the ...
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0answers
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Question on Galois Theory involving polynomials

Let $f(x) \in \mathbb{Q}[x]$ be irreducible and let $K$ be the extension of $\mathbb{Q}$ generated by all roots of $f(x)$ in $\mathbb{C}$. Then the degree of $f$ divides the order of ...
0
votes
1answer
58 views

Prove if F is infinite and a,b are separable over F then there exists an element c in F such that F(a,b) = F(a+cb).

I know this is true if F is infinite but don't know how to prove it. And is this still true if F is finite? I think The Primitive Element Theorem is the keypoint to prove this statement. Primitive ...
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0answers
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Multiplication in finite field - matrix representation

I have question regarding multiplication in Galois Field. I know that if we have e.g. $GF(2^m)$ and we have its normal or polynomial basis, we can find matrix representation of the multiplication, ...
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vote
1answer
44 views

Galois Group of an Inseparable Polynomial

The following question has arisen as part of my revision and I want to seek clarity on how it should be answered: If $f$ is a separable polynomial over a field $K$ and $L$ is its splitting field, ...
5
votes
2answers
41 views

Irreducibility of polynomials of the form $x^p - n$ over the cyclotomic field $Q(\zeta_p)$?

Is there a general procedure for showing that the polynomial $x^p - n$ is irreducible over the cyclotomic field $Q(\zeta_p)$? ($\zeta_p$ a primitive pth root of unity, and $n \in \mathbb{N}$. Maybe ...