Is $\Bbb Q[\sqrt2]$ cyclotomic? This overview of Galois Theory claims that a field extension of $F$ is cyclotomic if it's obtained by adjoining an $n$th root of _any_ element of $F$. Wikipedia claims you have to adjoin a root of _unity_ (it also says you can only cyclotomically extend $\Bbb Q$, not an arbitrary field). Which definition is correct? Are both in use? If Wiki's definition is right, what's the term for the one given in the other article?

I think the correct definition is, that for any field $K$, the extension $K(\zeta_n)$ is called a cyclotomic extension of $K$, for $\zeta_n$ being a root of unity of order $n$. The word "cyclotomic" is used in this way for many other definitions, like the $n$-th cyclotomic polynomial $x^n-1$, and so on.

On the other hand I have to admit, that the source you have given really says that $\mathbb{Q}(\sqrt{2})$ is a cyclotomic extension of $\mathbb{Q}$. I think, this is not consistent with most of the other "cyclotomic" definitions.