# What is the meaning of $K/F$ is a cyclic extension?

I have it that $K/F$ is a (finite) field extension, what is the definition of when $K/F$ is called cyclic ?

I heard it while I studied Galois theory and it was defined as

$K/F$ is called cyclic if $Gal(K/F)$ is a cyclic group

where the notation $Gal$ means that $K/F$ is also Galois.

Does, in general, it means $Aut(K/F)$ is cyclic, without the requirement that the extension is Galois ? (how it is defined in the literature/what is the convention ?)

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I've never seen the term used without the hypothesis that the extension is Galois. Calling $\mathbb Q(\sqrt[3]2)/\mathbb Q$ cyclic is strange! :P – Mariano Suárez-Alvarez Oct 11 '12 at 8:11
@MarianoSuárez-Alvarez - I guess you are right. Do you want to make this an answer so I can accept ? – Belgi Oct 11 '12 at 8:54
You can make the following definition. Let $\mathbb{K}$ be a field and $\mathbb{F}$ a subfield. We say that $\mathbb{K}/\mathbb{F}$ is a cyclic field extension if and only if $\mathbb{K}/\mathbb{F}$ is Galois and $\text{Gal}(\mathbb{K}/\mathbb{F})$ is a cyclic group. – Haskell Curry Jan 25 '13 at 7:18

An extension of fields $K/L$ is called Galois if its both separable and normal. An extension is called abelian if $K/L$ is Galois and the Galois group $\mathrm{Gal}(K/L)$ is abelian and cyclic if the Galois group is cyclic.