Units and ideal classes

Abstract

Units An algebraic integer e is said to be a unit if 1/e is an algebraic integer. This…is equivalent to the condition N e = ± 1. Indeed the conjugates of an algebraic integer are again algebraic integers, whence, if e is a unit, then N e and 1/ N e are rational integers and so ±1. Conversely if N e = ±1 then 1/e = ± N e/e which is clearly an algebraic integer. The set of all units form a group U under multiplication and the set of units in a number field K form a subgroup U K . Further, we see that if [α], [β] are principal ideals in K then we have [α] = [β] if and only if α/β ∈ U K . In general, we say that non-zero algebraic numbers α, β are associated if α/β ∈ U . The units in ℚ are plainly ±1 whence they are all the roots of unity in ℚ, that is, the solutions of an equation x l = 1 for some positive integer l . The units of the quadratic field K = ℚ(√ d ), where d ≠ 1 is a square-free integer, were discussed in Section 7.3. It was shown there that for the imaginary quadratic field K with d x 2 − 1 for D x 4 − 1 for D = −4 and of x 6 − 1 for D = −3 where D denotes the discriminant of the field, that is, D = 4 d for d ≡ 2, 3 (mod4) and D = d for d ≡ 1 (mod4).