Is it true that for all $2 \leq n \in \mathbb N$ we can find an abelian extension of the rationals with squarefree discriminant and degree n? Are there even infinitely many?
Some even degrees may be handled by cyclotomic fields but certainly not all. For quadratic numberfields there are infinitely many: $\mathbb Q(\sqrt{d})$, for squarefree $d>0$ and $d \equiv 1 \mod 4$.
I found this paper implying the existence of infinitely many extensions of squarefree discriminant and fixed degree n, but they all seem to have $S_n$ as galois group of their normal closure.