reference request - Looking for a proof that finite abelian groups are realizable over $mathbb{Q}$.

I've seen it claimed in a few places that all finite abelian groups are realizable over $\mathbb{Q}$, but I'm as yet unable to find a proof of this written down. Would it be possible for someone to point me towards a book that gives this result?

Answer

It's theorem 5.1 in Inverse Galois Theory by Malle and Matzat.