# Notation

\todo[inline]{Todo: definitions.}

- $U(\ZZ/k), (\ZZ/k)\units$ denotes the group of 
- $\ZZ/k$: the integers modulo $k$.
- $\bar{\ZZ}, \bar{\QQ}$: the algebraic closures of $\ZZ, \QQ$ respectively.
- $K, L$ will generally denote fields, and most likely number fields (so extensions of $\QQ$)
  $K/\QQ$ will denote that $K$ is a field extension of $\QQ$, and $K/L/\QQ$ denotes a tower of extensions $K/L$ and $L/\QQ$.
- $[K: \QQ]$ is the degree of the extension $K/\QQ$, i.e. the dimension of $K$ as a $\QQ\dash$vector space.
- $K(\alpha)$ will denote adjoining an element $\alpha \in \bar{K}$ to $K$ to get a field extension $K(\alpha)/K$.
- $\ZZ_K \da \bar{\ZZ} \intersect K$, the algebraic integers in $K$.
- $\ff(K)$ is the fraction field of $K$.
- $\Ab, \modsleft{\ZZ}$: the category of abelian groups.
- $\zeta_n$ will denote a primitive $n$th root of unity, so e.g. for $n$ prime one can take $\zeta_n \da e^{2\pi i \over n}$.
- $H\leq G$: $H$ is a subgroup of $G$.