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A monic polynomial 
$f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if 
$f(x)$ is irreducible over 
${\mathbb Q}$ and 
$\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of 
${\mathbb Q}(\theta )$, where 
$f(\theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form 
$x^4+bx^2+d$ 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.
