# 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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### minimal polynomial in Kummer extension

Let $n>1$ be an integer. Let $K$ be a field such that $n$ does not divide the characteristic of $K$ and $K$ contains the $n$-th roots of unity. Let $\mu_n\subseteq K$ be the set of $n$-th roots of ...
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### Structure and normality of the Galois group of $x^{15}-15 \in \mathbb{Q}[x]$.

Let $f(x) = x^{15}-15\in \mathbb{Q}[x]$. By Eisenstein's Criterion (using 3 or 5), $f$ is irreducible. Then $L=\mathbb{Q}(\sqrt[15]{15}, \omega)$ is the splitting field of $f$, where $\omega$ is a ...
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### Irreducible polynomial in finite-fields

Let $\overline{\mathbb{F}}_2$ be an algebraic closure of $\mathbb{F}_2$, and let $\alpha\in\overline{\mathbb{F}}_2$ be such that $\alpha^2+\alpha+1=0$. Prove that if a polynomial $P$ is irreducible in ...
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### Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit?

Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit? I know the definition of orbit and I know that $\sigma$ in Galois group changes roots. I ...
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### When do the lifting properties characterize covering spaces?

A covering map $p:E \rightarrow B$ is continuous map satisfying the following: For every point $b\in B$, there exists a neighborhood $U$ of $b$, such that $p^{-1}(U)$ is a disjoint union of open ...
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### Let $K_0$ denote the prime subfield of $K$ then $\text{Gal}(K/K_0)=\text{Aut}(K)$
Let $K_0$ denote the prime subfield of $K$ then $\text{Gal}(K/K_0)=\text{Aut}(K)$. This is a statement in the book I am looking at, it is given without proof or further explanation. I don't see ...
I wonder is there any easy way to evaluate elements of GF$(256)$: meaning that I would like to know what $\alpha^{32}$ or $\alpha^{200}$ is in polynomial form? I am assuming that the primitive ...