[Galois Theory] Is the converse of the Kronecker-Weber theorem true?

Isn't it obvious because a quotient of an abelian group can't be non-abelian?

Of course another case of violating "having an abelian Galois group" is not being a Galois extension to begin with... if you include those then of course it would be false (i.e. there exists algebraic number alpha such that Q(alpha) is not even Galois over Q, but some cyclotomic extension can contain alpha, when the splitting field of alpha is abelian over Q)
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