In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959) (岩澤 健吉), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives.
Formulation Iwasawa worked with so-called Z p {\displaystyle \mathbb {Z} _{p}} -extensions: infinite extensions of a number field F {\displaystyle F} with Galois group Γ {\displaystyle \Gamma } isomorphic to the additive group of p-adic integers for some prime p. (These were called Γ {\displaystyle \Gamma } -extensions in early papers.) Every closed subgroup of Γ {\displaystyle \Gamma } is of the form Γ p n , {\displaystyle \Gamma ^{p^{n}},} so by Galois theory, a Z p {\displaystyle \mathbb {Z} _{p}} -extension F ∞ / F {\displaystyle F_{\infty }/F} is the same thing as a tower of fields
F = F 0 ⊂ F 1 ⊂ F 2 ⊂ ⋯ ⊂ F ∞ {\displaystyle F=F_{0}\subset F_{1}\subset F_{2}\subset \cdots \subset F_{\infty }}
such that Gal ( F n / F ) ≅ Z / p n Z . {\displaystyle \operatorname {Gal} (F_{n}/F)\cong \mathbb {Z} /p^{n}\mathbb {Z} .} Iwasawa studied classical Galois modules over F n {\displaystyle F_{n}} by asking questions about the structure of modules over F ∞ . {\displaystyle F_{\infty }.}
More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.
Example Let p {\displaystyle p} be a prime number and let K = Q ( μ p ) {\displaystyle K=\mathbb {Q} (\mu _{p})} be the field generated over Q {\displaystyle \mathbb {Q} } by the p {\displaystyle p} th roots of unity. Iwasawa considered the following tower of number fields:
K = K 0 ⊂ K 1 ⊂ ⋯ ⊂ K ∞ , {\displaystyle K=K_{0}\subset K_{1}\subset \cdots \subset K_{\infty },}
where K n {\displaystyle K_{n}} is the field generated by adjoining to K {\displaystyle K} the pn+1-st roots of unity and
K ∞ = ⋃ K n . {\displaystyle K_{\infty }=\bigcup K_{n}.}
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