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If f is a degree n irreducible polynomial in Q[x] and the Galois group G of f is abelian, then

1. How do we know that G has exactly n elements?

2. Is the Galois group necessary cyclic?

I think that since f is irreducible, the Galois group must contain an automorphism of order n. So n [itex]\leq |Gal(f)| [/itex]. But what about the other inequality?

As for the answer to my second question, I thought it would be yes but now as I think about it, maybe not because G is a finitely generated abelian group. So by the Fundamental Theorem of Finitely Generated Abelian Groups, if [itex]n=(p_1)^{a_1} (p_2)^{a_2} ... (p_k)^{a_k} [/itex], then [itex]Gal(f) \cong \left[\frac{\mathbb{Z}}{p_1^{a_1}\mathbb{Z}} \times ... \times \frac{\mathbb{Z}}{p_k^{a_k} \mathbb{Z}} \right][/itex]????

Please help...