Number Fields and Number Rings

Abstract

A number field is a subfield of ~ having finite degree (dimension as a vector space) over II, 0 We know (see appendix 2) that every such field has the form Q[a] for some algebraic number a E ~ 0 If a is a root of an irreducible polynomial over II, having degree n, then Q[a] and representation in this form is unique; in other words, (l,a, 0 0 0 ,an -l) is a basis for Q[a] as a vector space over II, 0 We have already considered the field Q[w] where p prime. Recall that n = p -1 in that case. More generally, let 2ITi!m w = e } m not necessarily prime. The field Q[w] is called the mth cyclotomic field. Thus the first two cyclotomic fields are both just 11" since w = 1, -1 (resp.) for m = 1, 2. Moreover the third cyclotomic field is equal to the sixth: If we set w = e 2ITi / 6 , then w = _w 4 = _(w 2)2, which shows that Q[w] = Q[w 2 ] 0 In general, for odd m, the mth cyclotomic field is the same as the 2mth 0 (Show that if w = e 2TIi / 2m then w = _w m + l E Q[w 2 ] 0) On the other hand, We will show that the cyclotomic fields, for m even (m> 0), are all distinct. This will essentially follow from the fact (proved in this chapter) that the degree of the mth cyclotomic field over II, is ~(m), the number of elements in the set