Suppose I'm given a complex number expressible by radicals. I would like to determine whether that number belongs to a cyclotomic field.
Is there a computer algebra system (Magma?) in which I can test membership in a cyclotomic field? What function should I call to make this work?
The quickest way to check whether an algebraic number belongs to a cyclotomic field is probably to invoke Kronecker-Weber : an extension of $\mathbb Q$ is Galois abelian iff it's a subfield of a cyclotomic field.
So, given $\alpha$ in terms of nested radicals, compute its minimal polynomial, and then compute its Galois group. If it's abelian, then $\alpha$ belongs to a cyclotomic field.
SAGE is able to do all that.
(Funnily enough this method is non-constructive, in the sense that it doesn't actually tell you which cyclotomic field to look in!)