Let K be a number fieldMathworldPlanetmath with [K:]=n=r1+2r2. Here r1 denotes the number of real embeddings:


while r2 is half of the number of complex embeddings:


Note that {τj,τ¯j1jr2} are all the complex embeddings of K. Let r=r1+r2 and for 1ir define the “norm” in K corresponding to each embedding:


Let 𝒪K be the ring of integersMathworldPlanetmath of K. By Dirichlet’s unit theorem, we know that the rank of the unit group 𝒪K× is exactly r-1=r1+r2-1. Let


be a fundamental system of generators of 𝒪K× modulo roots of unityMathworldPlanetmath (this is, modulo the torsion subgroup). Let A be the r×(r-1) matrix


and let Ai be the (r-1)×(r-1) matrix obtained by deleting the i-th row from A, 1ir. It can be checked that the determinantMathworldPlanetmath of Ai, detAi, is independent up to sign of the choice of fundamental system of generators of 𝒪K× and is also independent of the choice of i.


The regulatorMathworldPlanetmath of K is defined to be


The regulator is one of the main ingredients in the analytic class number formulaMathworldPlanetmath for number fields.


  • 1 Daniel A. Marcus, Number Fields, Springer, New York.
  • 2 Serge Lang, Algebraic Number TheoryMathworldPlanetmath. Springer-Verlag, New York.
Title regulator
Canonical name Regulator
Date of creation 2013-03-22 13:54:34
Last modified on 2013-03-22 13:54:34
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 8
Author alozano (2414)
Entry type Definition
Classification msc 11R27
Related topic NumberField
Related topic DirichletsUnitTheorem
Related topic ClassNumberFormula
Related topic RegulatorOfAnEllipticCurve
Defines regulator of a number field