Let p be an odd prime number and let a be a root of the polynomial...

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Let p be an odd prime number and let a be a root of the polynomial m(X) = XP - 2p. (a) Show that m is irreducible over Q. (b) Calculate the discriminant of the basis {1, a, ..., ap-1}, simplifying your answer as much as possible. (c) Prove that Z[a] is the ring of algebraic integers in Q(a). (d) In the case p = 3, find all prime ideals of Z[a] with norm less than the Minkowski bound. (e) In the case p = 3, prove that Z[a] is a principal ideal domain. Let p be an odd prime number and let a be a root of the polynomial m(X) = XP - 2p. (a) Show that m is irreducible over Q. (b) Calculate the discriminant of the basis {1, a, ..., ap-1}, simplifying your answer as much as possible. (c) Prove that Z[a] is the ring of algebraic integers in Q(a). (d) In the case p = 3, find all prime ideals of Z[a] with norm less than the Minkowski bound. (e) In the case p = 3, prove that Z[a] is a principal ideal domain.