Monogenic Even Quartic Trinomials

A monic polynomial f(x)∈ Z[x] of degree N is called monogenic if $f(x)$ is irreducible over Q and \1,θ ,θ ²,… ,θ N-1\ is a basis for the ring of integers of Q(θ ) , where f(θ )=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.