6 - Units of rings of W-integers of norm 1

Summary

W-units play an important role in the construction of Diophantine definitions over number fields. In this chapter we discuss some properties of these units.

What are the units of the rings of W-integers?

Definition 6.1.1. Let K be a number field. Let W be a collection of its nonarchimedean primes. Let x ∈ K be such that its divisor is a product of powers of elements of W. Then x is called a W-unit. If W is empty then a W-unit is just an integral unit of K, i.e. an algebraic integer whose multiplicative inverse is also an algebraic integer.

Next we list some useful properties of these units. The first one is a generalization of the well-known Dirichlet unit theorem.

Proposition 6.1.2.Let K and W be as above. Then W-units form a multiplicative group. If the number of primes in W is finite, then the rank of this group is equal to the rank of the integral unit group plus the number of elements in W. (See the generalized version of the Dirichlet unit theorem in [64].)

The next lemma is a direct consequence of the definition of the rings of W-integers. (See Definition B.1.20.)

Lemma 6.1.3.Let K and W be again as above. Then the only invertible elements of O_{K, W} (or the only units of O_{K, W}) are the W-units.

The next lemma is important in making sure that the divisibility conditions discussed in Chapter 5 can be satisfied.