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[parent] index of the group of cyclotomic units in the full unit group (Theorem)

Let $ K_n=\mathbb{Q}(\zeta_{p^n})$ where $ \zeta_{p^n}$ is a primitive $ p^n$th root of unity, let $ h_n$ be the class number of $ K_n$ and let $ \mathcal{O}_n=\mathcal{O}_{K_n}$ be the ring of integers in $ K_n$. Let $ E_n=\mathcal{O}_n^\times$ be the group of units in $ K_n$. The cyclotomic units are a subgroup $ C_n$ of $ E_n$ which satisfy:

  • The elements of $ C_n$ are defined analytically.
  • The subgroup $ C_n$ is of finite index in $ E_n$. Furthermore, the index is $ h_{n}^+$: Let $ E^+_n$ be the group of units in $ K^+_n$ and let $ C^+_n=C_n\cap E^+_n$. Then $ [E_n^+:C_n^+]=h_{n}^+$. Moreover, it can be shown that $ [E_n:C_n]=[E_n^+:C_n^+]$ because $ E_n=\mu_{p^n}E_n^+$ (this is exercise 8.5 in [1]).
  • The subgroups $ C_n$ behave “well” in towers. More precisely, the norm of $ C_{n+1}$ down to $ K_{n}$ is $ C_{n}$. This follows from the fact that the norm of $ \zeta_{p^{n+1}}$ down to $ K_{n}$ is $ \zeta_{p^n}$.
Definition 1   Let $ p$ be prime and let $ n\geq 1$. Let $ \zeta_{p^n}$ be a primitive $ p^n$th root of unity.
  1. The cyclotomic unit group $ C_n^+\subset K_n^+=\mathbb{Q}(\zeta_{p^n})^+$ is the group of units generated by $ -1$ and the units
    $\displaystyle \xi_a=\zeta_{p^n}^{(1-a)/2}\frac{1-\zeta_{p^n}^a}{1-\zeta_{p^n}}=\pm \frac{\sin(\pi a/p^n)}{\sin(\pi/p^n)}$
    with $ 1<a<\frac{p^n}{2}$ and $ \gcd(a,p)=1$.
  2. The cyclotomic unit group $ C_n\subset K_n=\mathbb{Q}(\zeta_{p^n})$ is the group generated by $ \zeta_{p^n}$ and the cyclotomic units $ C_n^+$ of $ K_n^+$.
Remark 1   Let $ \sigma_a:\zeta_{p^n}\to \zeta_{p^n}^a$ be an element of $ \operatorname{Gal}(K_n/\mathbb{Q})$. Then:
$\displaystyle \xi_a=\zeta_{p^n}^{(1-a)/2}\frac{1-\zeta_{p^n}^a}{1-\zeta_{p^n}}=... ...ta_{p^n}^{-1/2}(1-\zeta_{p^n}))^{\sigma_a}}{\zeta_{p^n}^{-1/2}(1-\zeta_{p^n})}.$
Remark 2   Let $ g$ be a primitive root modulo $ p^n$. Let $ a\equiv g^r \mod p^n$. Then one can rewrite $ \xi_a$ as:
$\displaystyle \xi_a= \prod_{i=0}^{r-1}\xi_g^{\sigma_g^i}.$
In particular $ \xi_g$ generates $ C_n^+/\{\pm 1\}$ as a module over $ \mathbb{Z}[\operatorname{Gal}(\mathbb{Q}(\zeta_{p^n})^+/\mathbb{Q})]$.

Notice that in order to show that the index of $ C_n$ in $ K_n$ is finite it suffices to show that the index of $ C_n^+$ in $ K_n^+$ is finite. Indeed, let $ [K_n:\mathbb{Q}]=2d$. Since $ K_n$ is a totally imaginary field and by Dirichlet's unit theorem the free rank of $ E_n$ is $ r_1+r_2-1=d-1$. On the other hand, $ [K_n^+:\mathbb{Q}]=d$ and $ K_n^+$ is totally real, thus the free rank of $ E_n^+$ is also $ d-1$. Therefore the free rank of $ E_n^+$ and $ E_n$ are equal. As we claimed before, the index $ [E_n^+:C_n^+]$ is rather interesting to us.

Theorem 1 ([1],Thm. 8.2)   Let $ p$ be a prime and $ n\geq 1$. Let $ h^+_{n}$ be the class number of $ \mathbb{Q}(\zeta_{p^n})^+$. The cyclotomic units $ C_{n}^+$ of $ \mathbb{Q}(\zeta_{p^n})^+$ are a subgroup of finite index in the full unit group $ E_{n}^+$. Furthermore:
$\displaystyle h^+_{n}=[E_{n}^+:C^+_{n}]=[E_n:C_n].$
In the proof of the previous theorem one calculates the regulator of the units $ \xi_a$ in terms of values of Dirichlet L-functions with even characters. In particular, one calculates:
$\displaystyle R(\{\xi_a\})=\pm\prod_{\chi\neq \chi_0}\frac{1}{2}\tau(\chi)L(1,\overline{\chi})=h^+_{n}\cdot R^+$
where in the last equality one uses the properties of Gauss sums and the class number formula in terms of Dirichlet L-functions evaluated at $ s=1$. This yields that $ R(\{\xi_a\})$ in non-zero, therefore the index in $ E_n^+$ is finite and moreover
$\displaystyle h^+_{n}=\frac{R(\{\xi_a\})}{R^+}=[E_{n}^+:C^+_{n}]=[E_n:C_n].$
An immediate consequence of this is that if $ p$ divides $ h^+_{n}$ then there exists a cyclotomic unit $ \gamma \in C_n^+$ such that $ \gamma$ is a $ p$th power in $ E_{n}^+$ but not in $ C_n^+$.

Bibliography

1
L. C. Washington, Introduction to Cyclotomic Fields, Second Edition, Springer-Verlag, New York.



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Cross-references: divides, consequence, class number formula, Gauss sums, properties, equality, characters, even, terms, regulator, calculates, real, rank, Dirichlet's unit theorem, totally imaginary field, order, module, generates, primitive root, group generated by, units, generated by, group, prime, norm, index, finite, subgroup, cyclotomic units, group of units, ring of integers, class number, root of unity, primitive
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This is version 1 of index of the group of cyclotomic units in the full unit group, born on 2006-02-28.
Object id is 7661, canonical name is IndexOfTheGroupOfCyclotomicUnitsInTheFullUnitGroup.
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AMS MSC: 11R18 (Number theory :: Algebraic number theory: global fields :: Cyclotomic extensions)
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