# Tagged Questions

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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### 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 ...
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### 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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### 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?
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### 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 ...
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### 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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### 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: $F=\{x \in K| \sigma(x)=x \}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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. ...
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### $\text{deg}(f)$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $n=8$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...