the cyclotomic units are algebraic units
Let be a cyclotomic extension of with chosen to be minimal and let be the ring of integers (), recall that the cyclotomic units are the elements of the form
with and relatively prime to (where ). Here we prove that these elements are indeed algebraic units, i.e. .
Lemma 1.
Proof.
In order to prove the lemma, we will check that both and are algebraic integers, thus is a unit. Notice that it suffices to prove that is an algebraic integer, because the rest follows from interchanging the role of and .
Let be relatively prime to , thus are units in and we can find an integer such that:
Note also that it follows that . Moreover, using the equality of polynomials:
we get:
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 |