This note deals with a slight change of Sinnott's definition of circular units of an abelian field. The reason for this new definition giving a bigger group of units is as follows. Sinnott's formula for the index of circular units in the full group of units contains two factors which are difficult to handle: the index of the Sinnott module and (in the case of an imaginary field) an unknown power of 2. If we use the mentioned definition, the power of 2 in the corresponding formula can be easily described.
1997 Academic Press
1. DEFINITIONS
Let k be an abelian field (i.e., a finite Galois extension of rational numbers Q with an abelian Galois group). For convenience we suppose k to be contained in the complex field C. For any positive integer n let n=e 2?iÂn , K n =Q(n) be the n th cyclotomic field, and let k n =k & K n .
Sinnott (see [S, p. 201]) defined the group of circular units C of k as the intersection C=E & D, where E is the full group of units of k and D is the multiplicative subgroup of k _ generated by
where Z means the set of rational integers.
Let P be the set of all rational primes p 3 (mod 4) satisfyingp # k. We define the group of units C$ as the intersection C$=E & D$, where D$ is the multiplicative subgroup of k _ generated by D _ [-p ; p # P].