Theorem E (Theorem 7.1 in [47]).
The Iwasawa Algebra \(\Lambda(\Z_p)\) is isomorphic to the power series ring \(\mathscr{O} \llbracket T\rrbracket\text{,}\) where the isomorphism is induced via the map \(\gamma\mapsto1+T\text{.}\)
Section 3.1 Power Series and the Iwasawa Algebra
From this section onwards, we closely follow [47, Chapter 7]. Extensions \(K\) of \(\Q_p\) are called Local Fields. We denote the ring of integers of a local field \(K\) by \(\mathscr{O}_K\text{.}\) One particular example we will be dealing with is the ring of integers of the local field \(\Q_p(\chi(1),\chi(2),\ldots)\text{,}\) obtained by adjoining the values of a Dirichlet character \(\chi\) to \(\Q_p\text{.}\) Following [47, Chapter 7], we denote these rings of integers by \(\mathscr{O}_\chi=\Z_p[\chi(1),\chi(2),\ldots]\) for Dirichlet characters \(\chi\text{.}\) For notational convenience, when \(\chi\) is clear from the context, following [47, Chapter 7], we write \(\mathscr{O}\) instead of \(\mathscr{O}_\chi\text{.}\) Since such a ring \(\mathscr{O}\) is a complete DVR, it has a unique maximal ideal which is principal and we often denote that by \(\ideal{\pi}\) for a uniformizer \(\pi\text{.}\) For \(\mathscr{O}=\Z_p\text{,}\) we have \(\pi=p\text{,}\) i.e., the unique maximal ideal in \(\Z_p\) is \(p\Z_p\) and the quotient field \(\Z_p/p\Z_p\) is isomorphic to \(\F_p\text{.}\) Let \(\Gamma\) (in §2.2, we will consider a concrete example of such a group \(\Gamma\)) be a multiplicative group isomorphic to the additive group \(\Z_p\text{.}\) Since \(\Z_p\) is topologically generated by \(1\in\Z_p\text{,}\) i.e., the cyclic subgroup \(\ideal{1}=\Z\subset\Z_p\) is dense in \(\Z_p\text{,}\) \(\gamma\in\Gamma\text{,}\) the element corresponding to \(1\in\Z_p\) via the isomorphism, topologically generates \(\Gamma\text{.}\) The subgroups (closed balls of radius \(1/p^n\))
\begin{gather*}
p^n\Z_p=\{x\in\Z_p\colon\abs{x}_p\leq1/p^n\}
\end{gather*}
are the closed subgroups of \(\Z_p\text{.}\) Hence, the closed subgroups of \(\Gamma\) are of the form \(\Gamma^{p^n}=\{\lambda^{p^n}\colon\lambda\in\Gamma\}\text{.}\) Therefore, the isomorphism \(\Gamma\cong\Z_p\) induces isomorphism on the quotients \(\Gamma_n\colonequals\Gamma/\Gamma^{p^n}\cong\Z_p/p^n\Z_p\text{.}\) Consequently, \(\Gamma_n\) is cyclic of order \(p^n\text{.}\) Consider the group ring \(\mathscr{O}[\Gamma_n]\) for \(n\geq0\text{.}\) For \(m\geq n\geq 0\text{,}\) the natural map \(\Gamma_m\longrightarrow\Gamma_n\) induces a map of group rings \(\phi_{m,n}\colon\mathscr{O}[\Gamma_m]\longrightarrow\mathscr{O}[\Gamma_n]\text{.}\) Therefore, \(\{\phi_{m,n},\mathscr{O}[\Gamma_n]\}_{m\geq n\geq0}\) is an inverse system and we define
\begin{gather}
\Lambda(\Z_p)\colonequals\varprojlim_{n}\mathscr{O}[\Gamma_n].\tag{3.6}
\end{gather}
The algebra \(\Lambda(\Z_p)\) in definition (3.6) was first considered, for \(\mathscr{O}=\Z_p\text{,}\) by Iwasawa in his seminal 1959 paper [21]; hence it’s known as the Iwasawa Algebra. The following theorem structure theorem for the Iwasawa Algebra is crucial for this thesis, which we state without proof:
2
Note that [47] uses the notation \(\mathscr{O}\llbracket\Gamma\rrbracket\) instead of \(\Lambda(\Z_p)\text{.}\)
