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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3answers
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Why if $E$ has characteristic $p$ then $F=\{a\in E:a^{p^n}=a\}$ is a subfield?

We want to prove there exists a finite field of $p^n$ elements ($p$ is prime and $n>0$). Take $q=p^n$ and $g(x)=x^q-x\in\mathbb{Z}_p[x]$, and let $E$ be a field that contains $\mathbb{Z}_p$ and all ...
2
votes
2answers
44 views

Finding the degree of $\sqrt[3]{2} + \sqrt[3]{3}$ over $\mathbb Q$ [duplicate]

I am practicing writing down random algebraic numbers and finding their degrees over $\mathbb Q$ and have fumbled when coming to $\sqrt[3]{2} + \sqrt[3]{3}$. Mathematica tells me the minimal ...
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0answers
37 views

An Abelian group that is not solvable.

I'm aware that every finite abelian group is solvable, but is there an infinite abelian group that is not solvable?
2
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1answer
49 views

Solvable Groups

Does there exist a group $G$ such that every subgroup $H$ is solvable, but $G$ is not solvable. I know that if $G$ is solvable, then every subgroup $H$ is solvable, but I want to know if there is a ...
1
vote
1answer
24 views

Separable Extension and Splitting Field

Is every Separable extension a splitting field? Does there exist a counterexample? Also, is there an algebraically closed extension that is not separable?
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2answers
60 views

K is normal over F.

Let K be a field and suppose that $\sigma \in Aut(K)$ has infinite order. Let F be the fixed field of $\sigma$. If K/F is algebraic, show that K is normal over F. Note: $F=\{x \in K| \sigma(x)=x \}$ ...
3
votes
1answer
40 views

Understanding part of proof of the Fundamental Theorem of Galois Theory

I'm trying to understand the proof of the statement that, if $L\mathop{:}K$ is a finite,normal and separable with intermediate field $M$ such that $M\mathop:K$ is a normal extension, then the quotient ...
4
votes
2answers
69 views

Galois group of $\overline{\mathbb{F}_{p}}$ gives arithmetical information for finite fields $K/\mathbb{F}_{p}$?

Let $\mathbb{F}_{p}$ be the field with $p$ elements and $\overline{\mathbb{F}_{p}}$ be its algebraic closure. For some reason, we want to understand the structure of the Galois group of such an ...
0
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1answer
28 views

multiple of a crossed homomorphism from finite group to a divisible one is principal

Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed ...
3
votes
4answers
120 views

minimal polynomial $\zeta_n$ and $\zeta_n^p$ is the same for any prime $p$ not dividing $n$

I want to prove that for any prime $p$ not dividing $n$, $\zeta_n$ and $\zeta_n^p$ have the same minimal polynomial over $\mathbb{Q}$. My proposed proof, Suppose $\zeta_n$ is a primitive $n$th root ...
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0answers
22 views

Galois correspondence for $\mathbb{Q}(\sqrt[4]{2},i)/\mathbb{Q}$

I've determine that this extension has degree 8 and a basis of this extension is given by $$\{1, i, \sqrt[4]{2}, \sqrt[4]{2}i, \sqrt{2}, \sqrt{2}i, \sqrt[4]{8}, \sqrt[4]{8}i \}.$$ This reveals to us ...
2
votes
1answer
45 views

Nonabelian Galois Group

Let $f(x)$ be an irreducible polynomial in $\mathbb{Q}[x]$ with both real and nonreal roots. Show that its Galois group is nonabelian. Can the condition that $f$ is irreducible be dropped? If not, ...
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0answers
20 views

Using the discriminant to find Galois group of a quartic

I am working on finding the Galois groups of polynomials - in particular polynomials of degree $4$ I know that if we have a polynomial of the form $X^4+pX^2+qX+r$ we can find its cubic resolvent. ...
2
votes
1answer
32 views

$\text{deg}(f)$ is not divisible by $[L:F]$

I am trying to recall an exam question so I am sorry if this question doesn't make full sense. I think some people would know what the actual wording should be after reading it. $F \subseteq L$ is ...
0
votes
1answer
40 views

Splitting field of $x^3-5 \in \mathbb{Q}[X]$. Galois group and fields?

I have this multi-part problem I have worked on in Galois Theory. I am particularly unsure abut finding all roots of our polynomial and the action of the Galois group. Also, I cannot see how we can ...
0
votes
1answer
68 views

Prove there is an $A \in \mathbb{Q}$ such that $K(i)=\mathbb{Q}(i, \sqrt[4]{A})$

Let $K=\mathbb{Q}(\sqrt{-13+2\sqrt{13}})$. $K$ is normal over $\mathbb{Q}$ Prove there is an $A \in \mathbb{Q}$ such that $K(i)=\mathbb{Q}(i, \sqrt[4]{A})$ So we need to show that $K(i)=\mathbb{...
0
votes
1answer
14 views

Generators of $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ - what is $\tau(\zeta_n)$?

I am trying to understand the structure of $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})\simeq Z_n^*$ for $n \in \mathbb{Z}$ - it would be great if someone could me understand the generators in this group so ...
1
vote
1answer
35 views

Minimal field extension of $\mathbb{Q}(\sqrt[6]{2})$ over $\mathbb{Q}$ and Galois group

Find a minimal field extension $L$ of $\mathbb{Q}(\sqrt[6]{2})$ such that $L$ is normal over $\mathbb{Q}$ $L$ is normal over $\mathbb{Q}$ which means that it is the splitting field of polynomials in ...
1
vote
2answers
26 views

Existence of subfields such that $[\mathbb{Q}(\zeta_{25}) : K_1]=4$ and $[\mathbb{Q}(\zeta_{25}) : K_2]=5$

I have the following two questions questions I am working on and am a little stuck : Let $L=\mathbb{Q}(\zeta_{25})$ where $\zeta_{25}$ is the primitive $n$-th root of unity. Prove that there are ...
3
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2answers
65 views

Galois groups of $x^3-3x+1$ and $(x^3-2)(x^2+3)$ over $\mathbb{Q}$

I want to find the Galois groups of the following polynomials over $\mathbb{Q}$. The specific problems I am having is finding the roots of the first polynomial and dealing with a degree $6$ polynomial....
1
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1answer
18 views

Irreducible polynomial of degree $5$ in $L[X]$ using Kummer's theory

Let $L=\mathbb{F_2}(\theta)$ where $\theta$ is a root of $X^4+X+1$ I am trying to solve the following consecutive questions in Galois Theory: Prove that $L$ has degree $4$ over $\mathbb{F_2}$ ...
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2answers
51 views

Is $X^5+…+1 \in \mathbb{F_2}[X]$ irreducible?

I am trying to determine if the following polynomials are irreducible in $\mathbb{F_2}[X]$ are irreducible: $f(X)=X^5+X^2+1$ $g(X)=X^5+X^3+1$ There are no linear factors since $f(0)=f(1)=g(...
2
votes
2answers
29 views

Irreducibility of $X^5-7$ over $\mathbb{Q}(\sqrt[7]{2})[X]$ and degree of spitting field

I have worked through these two questions but am unsure if I got the right idea, please may you help me? Prove that $X^5-7$ is irreducible over $\mathbb{Q}(\sqrt[7]{2})[X]$ Can we say that $f(X)...
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0answers
31 views

Finding a $\gamma$ to define a Kummer extension like $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$

Previous theory: All the cyclic extensions of order $5$ are $\mathbb{Q}(\zeta_5)(\sqrt[5]{\gamma})/\mathbb{Q}(\zeta_5)$ where $\zeta_5$ is the generator of the group $\left(\mathbb{Z}/5\mathbb{Z}\...
0
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2answers
44 views

Non-separable, infinite field extensions of non-zero characteristic

I have been trying to find examples (and non-examples) of fields which are separable, finite and have characteristic equal to zero. Separable Example: $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ because the ...
2
votes
3answers
62 views

Describe the structure $\operatorname{Gal}(\mathbb{Q}(\zeta_4)/\mathbb{Q})$

I know that if $n$ is prime then $G=\operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \simeq \Bbb Z_{n-1}$ But I am unsure what $G$ is when $n$ is not prime. For example when $n=4$: $\operatorname{...
0
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0answers
29 views

Prove that $\phi_{2n}(X)=\phi_n(-X)$

Cyclotomic polynomials satisfy $x^n-1=\prod_{n_1|n}\phi_{n_1}(X)$ Prove that $\phi_{2n}(X)=\phi_n(-X)$ I need to be able to show that: $\zeta$ is an $n$-th primitive root of unity then $-\zeta$ ...
1
vote
1answer
30 views

Number of fields between $\mathbb{Q}$ and $\mathbb{Q}(\zeta_n)$

If $\zeta_n$ is the $n$-th primitive root of unity then $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \simeq Z_n^*$ due to the following map $$\tau(\zeta_n)=\zeta^n$$ I was wondering if we could use this ...
5
votes
2answers
51 views

Minimal polynomial of $\alpha$ over $\mathbb{Q}$ and $\mathbb{Q}(\sqrt{2})$

I have been working on these two minimal polynomial questions and am particularly concerned about (b) Find the minimal polynomial for $\sqrt[3]{4}+\sqrt[3]{2}$ (a) over $\mathbb{Q}$ By ...
0
votes
0answers
15 views

Existence of separable extensions of degree a power of the characteristic

Let $K$ be a field of characteristic $p$. For which $K$ there exist separable extensions of degree $p^n$ for every $n$? Attempt: If $K$ does not contain the algebraic closure of $\mathbb{F}_p$ the ...
0
votes
2answers
31 views

Transitive group not necessarily have an $n$-cycle

I know that it is true that if $G$ has an $n$-cycle, then $G$ is transitive as a subgroup of $S_n$, but now I'm trying to find an example why the converse is false. I've been trying examples with $...
0
votes
1answer
23 views

How are normal and separable extensions linked in Galois theory?

I am sure this is quite a basic question but I am having trouble understanding whole these key concepts are related Say we had a field extension $L/K$ .. How do the concepts of being normal and ...
1
vote
2answers
56 views

Prove that $K$ is normal over $\mathbb{Q}$ and $K(i)=\mathbb{Q}(i, \sqrt[4]{A})$

Let $K=\mathbb{Q}(\sqrt{-13+2\sqrt{13}})$ Prove that $K$ is normal over $\mathbb{Q}$ Need to show that $K$ is a splitting field of some polynomials in $\mathbb{Q}[X]$. Let $X=\sqrt{-13+2\sqrt{13}...
1
vote
1answer
32 views

Proving a polynomial splits over a certain field extension

If $K$ is the splitting field of $f\in F[x]$, and $g\in F[x]$ is irreducible and has a root in $K$, prove that $g$ splits over $K$. My proof (which I don't think is correct) is as follows: Let $a\in ...
1
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3answers
63 views

Constructible $n$-gons

Let $\xi$ be the primitive root of unity. If $n=5$, then the minimal polynomial of $xi$ over rationals would have degree $5-1=2^2$ which is a fermat prime so $5$-gon is constructible. If $n=8$, ...
2
votes
5answers
96 views

Prove that the $7$-th cyclotomic extension $\mathbb{Q}(\zeta_7)$ contains $\sqrt{-7}$

Prove that the $7$-th cyclotomic extension $\mathbb{Q}(\zeta_7)$ contains $\sqrt{-7}$ I thought that the definition of the $n$-th cyclotomic extension was: $\mathbb{Q}(\zeta_n)=\{\mathbb{Q}, \sqrt{-...
0
votes
1answer
21 views

Given $F \subset L \subset K$ where $K$ is a Galois ext. of $F$, find an example where $F \subset L$ is not a Galois ext.

I have already shown that if $F\subset K$ is a Galois extension, then for any intermediate field $L$, we have $L\subset K$ is a Galois extension. I then want to show that it's not necessarily true ...
0
votes
2answers
47 views

$\mathbb{Q}(\zeta_p)$ contains only one subfield $L$ such that $[L : \mathbb{Q}]=3$

Suppose $p$ is a prime number, $p\equiv1$ mod $3$ and $\mathbb{Q}(\zeta_p)$ is the $p$-th cyclotomic extension. Prove that $\mathbb{Q}(\zeta_p)$ contains only one subfield $L$ such that $[L : \...
2
votes
1answer
54 views

Is there a proof that this polynomial solvable by radicals?

If $f(x)$ is the minimal polynomial of $t$, a constructible number, over $\mathbb{Q}$, then is $f(x)$ solvable by radicals? It seems to be true, at least with the examples I came up with. Can it be ...
0
votes
1answer
18 views

Requesting material on Galois theory [duplicate]

Hello I wish to study Galois theory independently. I have no previous exposure to Galois theory specifically and was wondering if anyone had any tips or recommendations of good material I can use to ...
1
vote
1answer
24 views

minimal polynomial of $\alpha - \beta$ from the minimal polynomial of $\alpha$

If $f \in k[x]$ is the minimal polynomial of $\alpha \in K$, show how to write down the minimal polynomial of $\alpha - \beta$ for $\beta \in k$. I've been playing around with the minimal polynomial ...
0
votes
1answer
45 views

Minimal polynomial using Galois theory

I have a couple of questions, given below, about the following problem from a course in Galois Theory. Let $K=\mathbb{Q}(\zeta_{13})$. $K$ contains a unique subfield $L_4$ such that $[L_4 : \mathbb{Q}...
0
votes
1answer
15 views

Generator of $Gal(K/\mathbb{Q})$

Let $K=\mathbb{Q}(\zeta_5)$. Prove that there is a $\tau \in G$ such that $\tau \zeta_5=\zeta_5^2$ is a generator of $Gal(K/\mathbb{Q})$ I belive we must consider $\mathbb{Z_5}$, but I am not ...
0
votes
1answer
32 views

Galois subfields and subgroups

Find the minimal Galois extension $L$ of $\mathbb{Q}$ containing $\mathbb{Q}(\sqrt[4]{5})$ $L=\mathbb{Q}(\sqrt[4]{5}, i)$ is a splitting field of $X^4-5$ over $\mathbb{Q}$ Describe the structure ...
3
votes
2answers
40 views

I don't understand this argument about a certain Galois group.

So I'm working with $\alpha = \sqrt{5+\sqrt{5}}$ and $E=\mathbb{Q}(\alpha)$. The minimal polynomial of $\alpha$ over $\mathbb{Q}$ is $f(x) = x^4 -10x^2 +20$ and I've determined that $E$ is its ...
3
votes
0answers
45 views

Compute the Galois group of $p(x)=x^4+ax^3+bx^2+cx+d$

Compute the Galois group of the following polynomial: $$p(x)=x^4+ax^3+bx^2+cx+d$$ Step 1: Calculate cubic resolvent Step 2: Calculate the discrimimant $\gamma$ of the cubic resolvent Step 3: If $...
3
votes
2answers
57 views

Prove that $\mathbb{Q}$ has extensions of any finite degree in $\mathbb{C}$

This is a question from a course in Galois Theory and I am quite confused. In general, the degree of a field extension $E/F$ is the dimension of the vector space $E$. What would $E$ and $F$ be in ...
1
vote
2answers
28 views

Is $Q(2^{\frac14},\sqrt7)/Q(\sqrt2)$ normal?

I think it is because $(x^2-\sqrt2)(x^2-7)$ is a polynomial over $Q(\sqrt2)$ and $Q(2^{\frac14},\sqrt7)$ is the splitting field of this over $Q(\sqrt2)$ $\iff$ $Q(2^{\frac14},\sqrt7)$ is normal. So ...
2
votes
1answer
25 views

I want to show how many intermediate fields there are between $GF(3^{12})$ and $GF(3^4)$.

So $E=GF(3^{12})$ is an extension of $F=GF(3^4)$, and I wish to know how many intermediate fields $F \subsetneq K \subsetneq E$ there are. By a result in Escofier's Galois Theory I have that $G={\rm ...
0
votes
1answer
36 views

$G$ has normal subgroup of order 5

Let $L$ be the splitting field of $x^5-7$ over $Q$ and let $G=\text{Gal}(L/Q)$ (I) Prove that $G$ has a normal subgroup of order $5$ (II) Prove that $G$ has a subgroup of order $4$ that is not ...