 
Summary: A simple method for finding an
integral basis of a quartic field defined
by a trinomial x4
+ ax + b
S¸aban Alaca and Kenneth S. Williams 1
ABSTRACT. Let K be an algebraic number field of degree n. The
ring of integers of K is denoted by OK. Let P be a prime ideal of OK,
let p be a rational prime, and let (= 0) 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 OK. 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 (i {1, 2, . . . , 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 for each prime p we determine a
system of polynomial congruences modulo certain powers of p, which
is such that a pintegral basis of K can be given very simply in terms
