Monogenic Even Quartic Trinomials

Abstract A monic polynomial f (x) Zx of degree N is called monogenic if f (x) is irreducible over Q and \1, , ², , ^{N-1\} is a basis for the ring of integers of Q (), where f () =0. We prove that there exist exactly three distinct monogenic trinomials of the form x⁴+bx²+d whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.