Math 2.0

Number theory Definition

Iwasawa theory

Builds on Algebraic number fieldGalois extensionGalois groupp-adic integerIdeal class groupCyclotomic field

Definition

Let pp be a prime number, and let KK be an algebraic number field. Let ℤp\Z_p be the p-adic integers with their additive group and pp-adic topology. A ℤp\Z_p-extension of KK is a Galois extension K∞/KK_\infty/K inside a fixed algebraic closure such that the following holds. Its Galois group, with the Krull topology, is isomorphic to ℤp\Z_p as a topological group. For n≥0n \ge 0, the nn-th layer KnK_n is the fixed field of the closed subgroup corresponding to pnℤpp^n \Z_p. Then Gal⁡(Kn/K)≅ℤ/pnℤ\Gal(K_n/K) \cong \Z/p^n\Z and K∞=⋃nKnK_\infty = \bigcup_n K_n. The Iwasawa algebra is Λ=ℤp[ ⁣[T] ⁣]\Lambda = \Z_p[\![T]\!], the ring of formal power series over ℤp\Z_p. Fix a topological generator γ\gamma of Gal⁡(K∞/K)\Gal(K_\infty/K). Then Λ\Lambda acts on each compact ℤp\Z_p-module with a continuous action of Gal⁡(K∞/K)\Gal(K_\infty/K), with TT acting as γ−1\gamma - 1. Iwasawa theory is the study of such Λ\Lambda-modules attached to ℤp\Z_p-extensions. Let AnA_n be the pp-part of the ideal class group of KnK_n. The main such module is X=lim←⁡nAnX = \varprojlim_n A_n, the inverse limit under the norm maps.

Remarks

Iwasawa's theorem: there are integers λ≥0\lambda \ge 0, μ≥0\mu \ge 0 and ν\nu such that ∣An∣=pμpn+λn+ν\abs{A_n} = p^{\mu p^n + \lambda n + \nu} for all sufficiently large nn. For a finite abelian extension KK of ℚ\Q and its cyclotomic ℤp\Z_p-extension, Ferrero and Washington proved μ=0\mu = 0.

Example: let pp be odd. The Galois group of ⋃nℚ(ζpn)\bigcup_n \Q(\zeta_{p^n}) over ℚ\Q is ℤp×≅μp−1×(1+pℤp)\Z_p^\times \cong \mu_{p-1} \times (1 + p\Z_p). As a topological group, 1+pℤp1 + p\Z_p is isomorphic to ℤp\Z_p. So the fixed field of the factor μp−1\mu_{p-1} is a ℤp\Z_p-extension of ℚ\Q, the cyclotomic ℤp\Z_p-extension. Its layer K1K_1 is the unique subfield of degree pp of the cyclotomic field ℚ(ζp2)\Q(\zeta_{p^2}).

Non-example: for pp odd, ⋃nℚ(ζpn)\bigcup_n \Q(\zeta_{p^n}) itself is not a ℤp\Z_p-extension of ℚ\Q. Its Galois group ℤp×\Z_p^\times has the nontrivial finite subgroup μp−1\mu_{p-1}, while ℤp\Z_p has no nonzero element of finite order. It is, however, a ℤp\Z_p-extension of ℚ(ζp)\Q(\zeta_p).

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptCyclotomic field, concept, no evidence levelCyclotomic fieldconceptGalois extension, concept, no evidence levelGalois extensionconceptGalois group, concept, no evidence levelGalois groupconceptIdeal class group, concept, no evidence levelIdeal class groupconceptp-adic integer, concept, no evidence levelp-adic integerconceptIwasawa theory, definition, no evidence levelIwasawa theorydefinition
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  1. Algebraic number field concept
  2. Cyclotomic field concept
  3. Galois extension concept
  4. Galois group concept
  5. Ideal class group concept
  6. p-adic integer concept
  7. Iwasawa theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Iwasawa_theory
About this page
TagMRRQEEP3T8
URIm2://MRRQEEP3T8
Kinddefinition
WritingW0 stub
MSC11R23
DefinesIwasawa theory
UsesAlgebraic number field, Galois extension, Galois group, p-adic integer, Ideal class group, Cyclotomic field
Changed2 November 2026

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