TITLE: Cyclic extensions
QUESTION [8 upvotes]: Hi. Are there nice/simple examples of cyclic extensions $L/K$ (that is, Galois extensions with cyclic Galois group) for which $L$ cannot be written as $K(a)$ with $a^n\in K$?
Thanks.

REPLY [5 votes]: Take $\zeta = e^{2\pi i/p}$ for a prime number $p\equiv1$ (mod 3), e.g. $p=7$.
Then  $Q(\zeta+\bar\zeta)$ is a totally real cyclic Galois extension of $\mathbf{Q}$ of degree a multiple of 3, hence contains a cubic extension $L$ that is Galois with cyclic Galois group.
Being totally real it cannot be the splitting field of a polynomial of the
form $X^3-a$. (or use David Loeffler's argument above).
Dirichlet's theorem on primes in arithmetic progressions assures us that  we have infinitely many such examples over the base $\mathbf{Q}$.