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  "body": "\n## Definition\n\nLet $p$ be a prime number, and let $K$ be an [[algebraic number field|MWBDFS2KJ5]].\nLet $\\Z_p$ be the [[p-adic integers|V6TJAFZ4MD]] with their additive group and $p$-adic topology.\nA *$\\Z_p$-extension* of $K$ is a [[Galois extension|0QMZAQ6QQ4]] $K_\\infty/K$ inside a fixed algebraic closure such that the following holds.\nIts [[Galois group|0XTY86BZKY]], with the Krull topology, is isomorphic to $\\Z_p$ as a topological group.\nFor $n \\ge 0$, the $n$-th *layer* $K_n$ is the fixed field of the closed subgroup corresponding to $p^n \\Z_p$.\nThen $\\Gal(K_n/K) \\cong \\Z/p^n\\Z$ and $K_\\infty = \\bigcup_n K_n$.\nThe *Iwasawa algebra* is $\\Lambda = \\Z_p[\\![T]\\!]$, the ring of formal power series over $\\Z_p$.\nFix a topological generator $\\gamma$ of $\\Gal(K_\\infty/K)$.\nThen $\\Lambda$ acts on each compact $\\Z_p$-module with a continuous action of $\\Gal(K_\\infty/K)$, with $T$ acting as $\\gamma - 1$.\nIwasawa theory is the study of such $\\Lambda$-modules attached to $\\Z_p$-extensions.\nLet $A_n$ be the $p$-part of the [[ideal class group|WQ46BD3ARY]] of $K_n$.\nThe main such module is $X = \\varprojlim_n A_n$, the inverse limit under the norm maps.\n\n## Remarks\n\nIwasawa's theorem: there are integers $\\lambda \\ge 0$, $\\mu \\ge 0$ and $\\nu$ such that $\\abs{A_n} = p^{\\mu p^n + \\lambda n + \\nu}$ for all sufficiently large $n$.\nFor a finite abelian extension $K$ of $\\Q$ and its cyclotomic $\\Z_p$-extension, Ferrero and Washington proved $\\mu = 0$.\n\nExample: let $p$ be odd.\nThe Galois group of $\\bigcup_n \\Q(\\zeta_{p^n})$ over $\\Q$ is $\\Z_p^\\times \\cong \\mu_{p-1} \\times (1 + p\\Z_p)$.\nAs a topological group, $1 + p\\Z_p$ is isomorphic to $\\Z_p$.\nSo the fixed field of the factor $\\mu_{p-1}$ is a $\\Z_p$-extension of $\\Q$, the *cyclotomic* $\\Z_p$-extension.\nIts layer $K_1$ is the unique subfield of degree $p$ of the [[cyclotomic field|NN77VYSGXR]] $\\Q(\\zeta_{p^2})$.\n\nNon-example: for $p$ odd, $\\bigcup_n \\Q(\\zeta_{p^n})$ itself is not a $\\Z_p$-extension of $\\Q$.\nIts Galois group $\\Z_p^\\times$ has the nontrivial finite subgroup $\\mu_{p-1}$, while $\\Z_p$ has no nonzero element of finite order.\nIt is, however, a $\\Z_p$-extension of $\\Q(\\zeta_p)$.\n",
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