The ring of integers

Unless otherwise stated, throughout this book K and k will be algebraic number fields, even though some results hold more generally. The ring of integers of k (yet to be defined) will be denoted by o or ok; the ring of integers of K will be D or DK. What are the properties which one would like the integers of k to have? Some obvious ones are the following:-

1. ok is a commutative ring.

2. ok ∩ Q = Z, so that the integers which are rational are just the rational integers.

3. ok ⊗zQ = k, so that each α in k can be written as cβ where c is in Q and β is an integer in k.

4. If α is in, the algebraic closure of Q, the property that α is an integer only depends on a and not on the field in which we are working; in other words,

5. If α and α′ are conjugate over Q and α is an integer, then so is α′.

There is a largest subring of k satisfying these conditions, but no smallest one; so we shall choose ok to be the largest such subring. It follows from 1, 2 and 5 that if α is an integer then its monic irreducible polynomial over Q has coefficients in Z.