JP Journal of Algebra, Number Theory and Applications
Volume 3, Issue 3, Pages 477 - 505
(December 2003)
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A
SIMPLE METHOD FOR FINDING AN INTEGRAL BASIS OF A
QUARTIC FIELD DEFINED BY A TRINOMIAL x4
+ ax + b
Şaban Alaca (Canada) and Kenneth S. Williams (Canada)
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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 a
(¹
0) Î
K. If nP(a)
³
0, then a is called a P-integral
element of K, where nP(a)
denotes the exponent of P in the
prime ideal decomposition of aOK,
If a is P-integral
for each prime ideal P of K such
that P|pOK,
then a
is called a p-integral element of K.
Let {w1,
w2,
…, wn}
be a basis of K over Q, where
each wi(i
Î
{1, 2, …, n}) is a p-integral element of K. If
every p-integral element a
of K is given as a
= a1w1
+ a2w2
+
×××
+ anwn,
where ai
are p-integral
elements of Q, then ai
{w1,
w2,
…, wn}
is called a p-integral 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
p-integral basis of K can be given
very simply in terms of a simultaneous solution t
of the congruences. These congruences are then
put together to give a system of congruences in
terms of whose solution an integral basis for K
can be given. |
Keywords and phrases: quartic
field, p-integral
basis, integral basis, discriminant. |
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