Use a resolvent for A 3 to determine which of the following polynomials has a Galois group over Q isomorphic to A 3 . (1) X 3 − 5 X 2 + 2 X + 1; (2) X 3 − 2 X 2 − X + 1; (3) X 3 + 2 X + 1; (4) X 3 − 1; (5) ( X 2 − 2)( X − 1); (6) X 3 − 3 X 2 + 3 X − 1. When the Galois group is not isomorphic to A 3 , describe the group. (Note: Not all of the polynomials are irreducible. What can we say in those cases?)
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