# Questions tagged [galois-extensions]

For questions about Galois extensions fields. We say $L/K$ 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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### Simple argument that field automorphisms can extend

I've been learning about Galois theory and in a lot of proofs, an automorphism $\sigma\in Gal(L/K)$ is extended to $\hat{\sigma}\in Gal(M/K)$, when $K\subseteq L\subseteq M$ and $M/K$ and $L/K$ are ...
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### When are the extensions $L(S)/K$ and $L(S)/L$ totally ramified?

Let $K$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$. Let us consider the polynomial ring $R=K[x_1,x_2,...,x_n]$ in $n$-variables and $f_1, f_2, \cdots, f_m \in K[x_1, \cdots, x_n]$. ...
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### Two elements are not equal by using algebraic independent trick

Definition. $\bar{\Bbb Q}(S)$ denotes an algebraic closure of ${\Bbb Q}(S)$ in $\Bbb R$, that is, $\bar{\Bbb Q}(S)$ is the set of $x\in\Bbb R$ that are algebraic over $\Bbb Q(S).$ It seems that I ...
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### Computing a certain Galois group

I am trying to answer this question: Suppose $p$ is an odd prime and $K/\mathbb{Q}$ is the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of 1 in $\mathbb{C}$. (a) Show that $K$...
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I am trying to see how can I extend the automorphisms of the Galois extension $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$, for $d$ square-free, to automorphisms of $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ that fix \$\...