 
Summary: On Voronoi's method for finding an
integral basis of a cubic field
S¸aban Alaca and Kenneth S. Williams 1
ABSTRACT. We give a new proof of Voronoi's determination of an
integral basis for a cubic field.
Let K be a cubic field. Without loss of generality we may take the
cubic field K in the form K = Q(), where is a root of the irreducible
polynomial
f(x) = x3
 ax + b , a, b Z.
For each prime p and each nonzero integer m, p(m) denotes the greatest
exponent l such that pl
 m. We can also assume that for every prime p
p(a) < 2 or p(b) < 3 ,
see [4, p. 579]. The discriminant of is = 4a3
27b2
and = i()2
d(K),
where i() denotes the index of and d(K) denotes the discriminant of K.
For each prime p, set sp = p( ) and p = /psp
