# 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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### Can we describe nicely all the rational numbers of the form $x^{2}-xy+y^{2}$?

Let $\zeta$ be a primitive cubic root of unity, i.e., $\zeta$ is a complex number such that $\zeta^{3}=1$ and $\zeta\neq1$. Let us consider the Galois extension $\mathbb{Q}(\zeta)/\mathbb{Q}$ (the ...
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### Counter-examples of Galois Correspondence

What are some examples of a separable field extension $L/K$ and a subgroup $H$ of $\text{Aut}(L/K)$ such that $\text{Aut}(L/L^H) \neq H$? Here $L^H$ means the fixed field of $H$.
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### Proof Enquiry, Field of order $p^n$ [duplicate]

I want to prove that there exists an inclusion $\mathbb{F}_{p^a} \hookrightarrow \mathbb{F}_{p^b}$ iff $a \vert b$. Suppose that $a \vert b$, then $b =ac$ for some $c \in \mathbb{Z}$. Consider then ...
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### Cardinality of Galois groups

We know that no Galois group of a Galois extension is countable. The question is: which cardinalities are possible for a Galois group? (or also: for profinite groups?) I suspect that the theory of ...
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### No $p$-th root implies $X^{p^n}-a$ irreducible for all $n \in \mathbb{N}$

While doing exercises of Chapter IV in Lang's algebra, I encountered the following problem: Suppose char $K=p$. Let $a \in K$. If $a$ has no $p$-th root in $K$, show that $X^{p^n}-a$ is ...