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# Questions tagged [galois-theory]

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-connections).

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### On the definition of algebraic closure

Let $F$ be a field. By definition, the following are equivalent: $F$ is algebraically closed. Every nonconstant polynomial in $F[x]$ splits over $F$. Every nonconstant polynomial in $F[x]$ has a ...
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Assume $K, L$ are $2^k, 2^l$-th cyclotomic field, respectively, that is, $K = \mathbb{Q}[x]/(x^{2^{k-1}}+1), L = \mathbb{Q}[x]/(x^{2^{l-1}}+1), l > k > 2$. Then $Gal(K /\mathbb{Q}) \cong \mathbb{... 0 votes 0 answers 109 views ### Why are the algorithms for finding the Galois group of a polynomial generally flawed? In Galois theory, the endeavor of algorithmically finding the Galois group of a polynomial$f$over the field$\mathbb{Q}$, where$f\in\mathbb{Q}[z]$is generally not well-accomplished. By this, i ... 1 vote 0 answers 26 views ### A question from Dummit-Foote on showing nontrivial algebraic extensions of a certain field (Section 14.9, Q.15) I am trying to solve a problem from Dummit-Foote's Section 14.9 (problem 15): Let$K_0= \mathbb Q$and for$n > 0$define the field$K_{n+1}$as the extension of$K_n$obtained by adjoining to$...
Suppose $p,q$ are distinct primes. $K_1$ is a Galois extension of $\mathbb{Q}(\mu_{pq})$ of degree $p$, $K_1 \subseteq \mathbb{C}$. $K_2$ is a Galois extension of $\mathbb{Q}(\mu_{pq})$ of degree $q$, ...