 
Summary: pIntegral bases of a quartic field
defined by a trinomial x4
+ ax + b
S¸aban Alaca and Kenneth S. Williams 1
ABSTRACT. Let P be a prime ideal of an algebraic number field K, let p be a
rational prime, and let K. If P () 0 then is called a Pintegral element
of K, where P () denotes the exponent of P in the prime ideal decomposition
of < >. If is Pintegral for each prime ideal P of K such that P  pOK then
is called a pintegral element of K. Let {1, 2,..., n} be a basis of K over Q,
where each i (1 i n) is a pintegral element of K. If every pintegral element
of K is given as = a11 + a22 + · · · + ann, where the ai are pintegral
elements of Q, then {1, 2,..., n} is called a pintegral basis of K. In this paper
a pintegral basis of a quartic field K defined by a trinomial is determined for
each rational prime p, and then the discriminant of K and an integral basis of K
are obtained from its pintegral bases.
1 Introduction
In this paper we determine for each prime p a pintegral basis for a quartic
field K = Q(), where is a root of the irreducible quartic trinomial x4
+
ax + b, a, b Z. The discriminant of K and an integral basis of K are then
