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  "body": "\n## Summary\n\nIwasawa theory studies arithmetic objects, such as [[ideal class groups|WQ46BD3ARY]], along an infinite tower of number fields $K = K_0 \\subset K_1 \\subset K_2 \\subset \\cdots$.\nThe union of the tower has Galois group isomorphic to the additive group of [[p-adic integers|V6TJAFZ4MD]] $\\Z_p$.\nSuch a tower is a $\\Z_p$-extension, and the main example is obtained from the fields $\\Q(\\zeta_{p^n})$ of $p$-power roots of unity.\nThe objects attached to the whole tower are modules over the Iwasawa algebra $\\Z_p[\\![T]\\!]$, whose structure theory gives regular growth formulas.\nThe [[Iwasawa main conjecture|67A4D4FAYJ]] links these modules to $p$-adic $L$-functions.\n\n## Explanations\n\n### L1 human\n\nInstead of studying one number field at a time, Iwasawa theory looks at an infinite tower of fields, each of degree $p$ over the one below.\nThe class numbers along the tower grow in a very regular way.\nIwasawa proved that the power of $p$ dividing the $n$-th class number is $p^{e_n}$ with $e_n = \\mu p^n + \\lambda n + \\nu$ for all large $n$, for some fixed integers $\\lambda, \\mu, \\nu$.\nFor example, if $p$ divides no class number in the tower $\\Q(\\zeta_{p^{n+1}})$, all three of these numbers are $0$.\nStudying the whole tower at once reveals structure that is hidden in each single field.\n",
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