6320-2024-08

Prove that the alternating group \(A_5\) is the only simple group of order 60.

Prove that if \(p\) is prime, then each group with \(p^d\) elements has a non-trivial center.

Let \(K\) be an arbitrary field. Prove that the polynomial ring \(K[x]\) has infinitely many prime ideals.

Show that if \(L/K\) is Galois and \(f \in K[x]\) is monic irreducible, then every irreducible factor of \(f\) in \(L[x]\) has the same degree.

Find a complete list of irreducible representations of the alternating group \(A_4\).