Prime that splits but not completely in $\mathbb{Q}(\zeta_{20})/\mathbb{Q}$

Question

I need a prime that splits but not completely in $\mathbb{Q}/(\zeta_{20})\mathbb{Q}$. Is there any way to find one quickly?

Answer 1

You can even do it in the sub-extension $L= \mathbb Q (\zeta_5)$: a prime $p$ that is congruent to $4 \pmod 5$ (e.g. $19$) splits completely in the quadratic sub-extension $K= \mathbb Q(\sqrt{5})$, but not in $L$.

For a quick sketch as to why this works, please see my answer Intuition of Prime decomposition in Galois extensions of subgroups of a cyclotomic polynomial.

Edit - any prime, the residue of which generates a (proper - but that is automatic) non-trivial subgroup of $(\mathbb Z/20)^*$, will work.