Abstract
Abstract
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.
Publisher
Cambridge University Press (CUP)