# 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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### Any direct method to show that $\mathbb{Q}[\sqrt{2}, \sqrt{3}, \sqrt{5}]=\mathbb{Q}[\sqrt{2}+\sqrt{3}+\sqrt{5}]$?

We know that Galois extension is simple extension, so the splitting field of $(X^2-2)(X^2-3)(X^2-5)$ over $\mathbb{Q}$ satisfies $\mathbb{Q}[\sqrt{2}, \sqrt{3}, \sqrt{5}]=\mathbb{Q}[\alpha]$, for some ...
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### non-isomorphic number fields

Show that following 2 polynomial do not generate isomorphic number number fields: $P_1(x)= x^3+10x+1$ $P_2(x) = x^3-8x+15$ I see $Disc(P_1)=Disc(P_2)=-4027$ and both of them have a real root and 2 ...
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### Sum of nth power of some of the roots of irreducible polynomial over $\mathbb{Q}$ is in $\mathbb{Q}$

So i know that for a splitting field K over $\mathbb{Q}$ of the polynomial f(x), where a,b,c,d are the roots of f(x). Taking the following sum $a^n +b^n+c^n +d^n$ is in the FixGal(K,$\mathbb{Q}$) ....
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### Does the Galois group of a polynomial change upon translation?

I know that a polynomial $f(X)$ is irreducible iff $f(X+1)$ is irreducible. Is it true for the Galois group? I think it should but I don't know if there ir a neat proof (I have thought of writing the ...
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### Lemma 13, Section 5.2 of Hungerford’s Algebra

Theorem 2.2. Let $F$ be an extension field of $K$ and $f\in K[x]$. If $u\in F$ is a root of $f$ and $\sigma \in \text{Aut}_K F$, then $\sigma (u)\in F$ is also a root of $f$. If $F$ is an extension ...
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### $f\in\mathbb{Q}[x]$ is irreducible in $\mathbb{Q}[x]$ and the Galois group of $f$ has order 99 , what is the degree of $f$.

I need help for the following problem(what is the key-idea that problem): Problem: Let $f(x)$ be a monic polynomial with rational coefficients. Assume $f(x)$ is irreducible in $\mathbb{Q}[x]$ and the ...
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### $Gal(LM/K)$ isomorphic to $\{ (\sigma , \tau) \in Gal(L/K) \times Gal(M/K) : \sigma |_{L \cap M} = \tau |_{L \cap M} \}$

Let L/K, M/K be finite Galois extensions contained in some common field extension $\mathbb{K}$ of $K$, so we can speak about $L \cap M$ and the composite $LM$ , which is defined to be the smallest ...
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### Some exercises about Galois Groups and Automorphisms of a finite field extension

Hey I have these two exercises where I am having some problems solving them: Let $E/K$ be a finite field extension, let $G$ be a subgroup of $Aut(E/K)$. Show: a) $G$ is finite and $|G|$ is a divisor ...
I have this two exercise where I have some problems: (a) We consider the finite field extension $E := \mathbb{Q}( \sqrt[4]{2}, i)$ over $\mathbb{Q}$. Show that $E$ is a Galois extension of $\mathbb{Q}$...