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[parent] the cyclotomic units are algebraic units (Theorem)

Let $ L=\mathbb{Q}(\zeta_m)$ be a cyclotomic extension of $ \mathbb{Q}$ with $ m$ chosen to be minimal and let $ \mathcal{O}_L$ be the ring of integers ( $ =\mathbb{Z}(\zeta_m)$), recall that the cyclotomic units are the elements of the form

$\displaystyle \eta=\frac{\zeta^r-1}{\zeta^s-1}$    

with $ r$ and $ s$ relatively prime to $ m$ (where $ \zeta=\zeta_m$). Here we prove that these elements are indeed algebraic units, i.e. $ \eta \in \mathcal{O}_L^\times$.
Lemma 1   The cyclotomic units are algebraic units.
Proof. In order to prove the lemma, we will check that both $ \eta$ and $ \eta^{-1}$ are algebraic integers, thus $ \eta$ is a unit. Notice that it suffices to prove that $ \eta$ is an algebraic integer, because the rest follows from interchanging the role of $ r$ and $ s$.

Let $ r,s\in \mathbb{Z}$ be relatively prime to $ m$, thus $ r \mod m, s\mod m$ are units in $ \mathbb{Z}/m\mathbb{Z}$ and we can find an integer $ a$ such that:

$\displaystyle a\cdot s \equiv r \mod m$
Note also that it follows that $ \zeta^r=\zeta^{as}$. Moreover, using the equality of polynomials:
$\displaystyle x^{as}-1=(x^s-1)\cdot(x^{s(a-1)}+x^{s(a-2)}+\ldots+x^s+1)$
we get:
$\displaystyle \eta$ $\displaystyle =$ $\displaystyle \frac{\zeta^r-1}{\zeta^s-1}=\frac{\zeta^{as}-1}{\zeta^s-1}$  
  $\displaystyle =$ $\displaystyle \frac{(\zeta^s-1)\cdot(\zeta^{s(a-1)}+\zeta^{s(a-2)}+\ldots+\zeta^s+1)}{\zeta^s-1}$  
  $\displaystyle =$ $\displaystyle \zeta^{s(a-1)}+\zeta^{s(a-2)}+\ldots+\zeta^s+1 \in \mathcal{O}_L=\mathbb{Z}[\zeta]$  

Hence the result. $ \qedsymbol$



"the cyclotomic units are algebraic units" is owned by alozano.
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Cross-references: polynomials, equality, integer, unit, algebraic integers, order, algebraic units, relatively prime, cyclotomic units, ring of integers, minimal, cyclotomic extension
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This is version 1 of the cyclotomic units are algebraic units, born on 2004-02-29.
Object id is 5657, canonical name is CyclotomicUnitsAreAlgebraicUnits.
Accessed 1455 times total.

Classification:
AMS MSC: 11R18 (Number theory :: Algebraic number theory: global fields :: Cyclotomic extensions)
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