the cyclotomic units are algebraic units

Let L=(ζm) be a cyclotomic extension of with m chosen to be minimal and let 𝒪L be the ring of integersMathworldPlanetmath (=(ζm)), recall that the cyclotomic units are the elements of the form


with r and s relatively prime to m (where ζ=ζm). Here we prove that these elements are indeed algebraic units, i.e. η𝒪L×.


In order to prove the lemma, we will check that both η and η-1 are algebraic integersMathworldPlanetmath, thus η is a unit. Notice that it suffices to prove that η is an algebraic integer, because the rest follows from interchanging the role of r and s.

Let r,s be relatively prime to m, thus rmodm,smodm are units in /m and we can find an integer a such that:


Note also that it follows that ζr=ζas. Moreover, using the equality of polynomialsPlanetmathPlanetmath:


we get:

η = ζr-1ζs-1=ζas-1ζs-1
= (ζs-1)(ζs(a-1)+ζs(a-2)++ζs+1)ζs-1
= ζs(a-1)+ζs(a-2)++ζs+1𝒪L=[ζ]

Hence the result. ∎

Title the cyclotomic units are algebraic units
Canonical name TheCyclotomicUnitsAreAlgebraicUnits
Date of creation 2013-03-22 14:13:17
Last modified on 2013-03-22 14:13:17
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Theorem
Classification msc 11R18
Related topic AlgebraicInteger