# Questions tagged [galois-extensions]

For questions about Galois extensions of fields. We say that an algebraic extension $L/K$ is a Galois extension iff the subfield of $L$ that is fixed by automorphisms of $L$ which fix K is exactly $K$.

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### Splitting field of a separable (and irreducible) polynomial is separable

I was struggling proving this and didn't manage to find any solution here that felt understandable for my level, so I am submitting my best idea, which seems right. Let $L$ be a splitting field of a ...
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### How to generate all elements of an extension field from a base field? (GF(2))

I am trying to understand how to show something is a primitive polynomial; I understand it has to be irreducible by definition, and according to Wolfram: ...
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### Action of Galois group on scheme

The following exercise is the 2.10 page 97 of Liu's book. Let $k/K$ a finite Galois extension and $X$ a $k-$scheme of finite type over $k$ (i.e. a $k$ algebraic variety). I have the following ...
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### Elements with norm $1$ in a quadratic field [closed]

Consider the number field $K=\mathbb{Q}(\sqrt{5})$ and the norm map $N:\mathbb{Q}(\sqrt{5}) \to \mathbb{Q}^*$ given by $N(a+b \sqrt{5})=a^2-5b^2$. Is there an explicit description of the kernel of ...
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### Determine the degree of a root of $X^p-X-\alpha$ (Serge Lang Algebra exercise VI.29)

Let $K$ be a cyclic extension of a field $F$, with Galois group $G=\langle \sigma \rangle$ and assume that $\operatorname{char}F=p$ and that $[K:F]=p^{m-1}$ for some $m>1$. Let $\beta$ be an ...
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### Generators of a Galois group in SAGEMath

I have a (potentially silly) question about a problem I've been having with SAGEMath. Let $K$ be the maximal totally real subfield of the cyclotomic field of conductor $24$, i.e. the number field with ...
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### Pove that the Galois group of $p(x) = f(x^2)$ is not abelian

Let $f(x) \in \mathbb Q[x]$ be an irreducible cubic polynomial with three real roots $\alpha, \beta, \gamma \in \mathbb R$ such that $\alpha$ is negative and $\beta$ is positive. Prove that the Galois ...
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### Example of a field on which every irreducible polynomial has degree a power of $p$

Exercise A-47 in Milne's Fields and Galois Theory notes asks to prove that if $p$ is a prime number and $F$ is a field of characteristic zero such that every irreducible polynomial $f(X)\in F[X]$ has ...
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### Compute $[ \mathbb Q(\zeta_8) : \mathbb Q(i)]$ and find a basis for the extension.

This is a follow-up to my previous question; I wanted to give a different example to make sure I understand what is going on. Compute $[ \mathbb Q(\zeta_8) : \mathbb Q(i)]$ and find a basis for the ...
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### Compute degree of $\mathbb Q(\zeta_8)$ over $\mathbb Q(\sqrt 2)$ and find a basis.

This is a review problem for a Galois theory course. Compute $[\mathbb Q(\zeta_8): \mathbb Q(\sqrt 2)]$ and find a basis. (Where $\zeta_8$ is the primitive $8$th root of unity). I compute the degree ...
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### Galois group of $x^4 +3 \in \mathbb Q[x]$ using Kaplansky's Theorem

This is exercise 4.5 from Baker Galois Theory: Use Kaplansky's theorem to find the Galois group of the splitting field $E$ of the polynomial $x^4 +3 \in \mathbb Q[x]$ over $\mathbb Q$. Determine all ...
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### Product vs. join of subgroups of a finite group

Hypotheses: $G$ is a finite group $A$ and $B$ are subgroups if $G$. Definitions: $AB = \{ ab \mathbin{|} a \in A, b \in B \}$ is the product of $A$ and $B$ $A \vee B$ is the join of (subgroup of $G$...
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### I am proposing two conjectures regarding the BCH codes ! Can anybody prove or disprove my conjectures?

I was working with BCH codes for Compressed Sensing. Through the simulation of BCH codes, I found some interesting facts which are useful for my Ph.D. thesis. I don't have mathematical proof of my ...
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### Galois group of $x^n+x+1$

I’m struggling with a following question. Having a polynomial $P(x) = x^{10} + x + 1 \in \mathbb{Q}[x]$ find its Galois group. The question for $x^{3} + x + 1$ is much easier for me, because, all the ...
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### Proving there are no intermediate normal extensions

Let $E/F$ be a finite extension and $L \supseteq E$ an splitting field of a polynomial $g \in F[x]$ such that every irreducible factor of $g$ in $F[X]$ has a root in $E$. I need to prove $L/F$ is a ...
### Why isn't the crossed product algebra $(K(\sqrt{a}), Gal(K(\sqrt{a})/K, k)$ independent of a?
I am trying to work through an example to understand the correspondence between $H_2(Gal(K(\sqrt{a})/K),K(\sqrt{a}))$ and $Br(K(\sqrt{a}),K)$. Here is my confusion: Let $E = K(\sqrt{a})$ be a ...