# index of the group of cyclotomic units in the full unit group

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. 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

 $\xi_{a}=\zeta_{p^{n}}^{(1-a)/2}\frac{1-\zeta_{p^{n}}^{a}}{1-\zeta_{p^{n}}}=\pm% \frac{\sin(\pi a/p^{n})}{\sin(\pi/p^{n})}$

with $1 and $\gcd(a,p)=1$.

2. 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:

 $\xi_{a}=\zeta_{p^{n}}^{(1-a)/2}\frac{1-\zeta_{p^{n}}^{a}}{1-\zeta_{p^{n}}}=% \frac{(\zeta_{p^{n}}^{-1/2}(1-\zeta_{p^{n}}))^{\sigma_{a}}}{\zeta_{p^{n}}^{-1/% 2}(1-\zeta_{p^{n}})}.$
###### 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:

 $\xi_{a}=\prod_{i=0}^{r-1}\xi_{g}^{\sigma_{g}^{i}}.$

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:

 $h^{+}_{n}=[E_{n}^{+}:C^{+}_{n}]=[E_{n}:C_{n}].$

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:

 $R(\{\xi_{a}\})=\pm\prod_{\chi\neq\chi_{0}}\frac{1}{2}\tau(\chi)L(1,\overline{% \chi})=h^{+}_{n}\cdot R^{+}$

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

 $h^{+}_{n}=\frac{R(\{\xi_{a}\})}{R^{+}}=[E_{n}^{+}:C^{+}_{n}]=[E_{n}:C_{n}].$

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 $C_{n}^{+}$.

## References

• 1 L. C. Washington, Introduction to Cyclotomic Fields, Second Edition, Springer-Verlag, New York.
Title index of the group of cyclotomic units in the full unit group IndexOfTheGroupOfCyclotomicUnitsInTheFullUnitGroup 2013-03-22 15:42:49 2013-03-22 15:42:49 alozano (2414) alozano (2414) 4 alozano (2414) Theorem msc 11R18