From Dummit and Foote I was reading about cyclotomic extensions, where I came across the definition of algebraic extension saying the extension $K/F$ is an algebraic extension if $K/F$ is Galois and $Gal(K/F)$ is a Galois group. But as we know every cyclotomic extensions give rise to an abelian group. Then does that mean there are abelian group may give two different extensions. It will be great if you can help me to clear this doubt. Thanks.
2026-03-25 01:26:01.1774401961
Question regarding Galois group and algebraic extensions
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Why two different extensions? The question is not clear. The definition of an algebraic extension has nothing to do with an abelian Galois group. A cyclotomic extension $\Bbb Q(\zeta_n)\mid \Bbb Q$ clearly is an algebraic extension of degree $\phi(n)$. It does not matter whether or not its Galois group is abelian or not (yes, it is abelian for cyclotomic extensions).