Math 2.0

Number theory Definition

Class number

Builds on Algebraic number fieldIdeal class groupOrder of a groupFiniteness of the class numberThe Gaussian integers are a Euclidean domainDedekind domainStark–Heegner theorem

Definition

Let KK be an algebraic number field and Cl⁡(K)\operatorname{Cl}(K) its ideal class group. The class number of KK is the order of this group: hK=#Cl⁡(K).h_K = \#\operatorname{Cl}(K). It is a positive integer by the finiteness of the class number.

Remarks

Example: for K=ℚ(i)K = \Q(i), the ring of integers ℤ[i]\Z[i] is a Euclidean domain, hence a principal ideal domain, so Cl⁡(K)\operatorname{Cl}(K) is trivial and hK=1h_K = 1. Example: for K=ℚ(−5)K = \Q(\sqrt{-5}), Cl⁡(K)≅ℤ/2ℤ\operatorname{Cl}(K) \cong \Z/2\Z, so hK=2h_K = 2. Non-example: for a general Dedekind domain the class group can be infinite, and then there is no finite class number. For instance, let EE be the elliptic curve y2=x3−xy^2 = x^3 - x over ℂ\C. The class group of its affine coordinate ring ℂ[x,y]/(y2−x3+x)\C[x, y]/(y^2 - x^3 + x) is isomorphic to the infinite group E(ℂ)E(\C). By the Stark–Heegner theorem, the imaginary quadratic fields ℚ(d)\Q(\sqrt{d}) with h=1h = 1 are those with d=−1,−2,−3,−7,−11,−19,−43,−67,−163d = -1, -2, -3, -7, -11, -19, -43, -67, -163.

Reading path

To understand this page from the ground up, read these 36 pages in order. The foundations come first.

Show the path as a graph
Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptBinary quadratic form, concept, no evidence levelBinary quadratic formconceptClass number, concept, no evidence levelClass numberconceptComplex multiplication, concept, no evidence levelComplex multiplicationconceptDedekind domain, concept, no evidence levelDedekind domainconceptDedekind zeta function, concept, no evidence levelDedekind zeta functionconceptDirichlet L-function, concept, no evidence levelDirichlet L-functionconceptDiscriminant of a number field, concept, no evidence levelDiscriminant of a num…conceptFree abelian group, concept, no evidence levelFree abelian groupconceptGroup, concept, no evidence levelGroupconceptGroup of units of a ring, concept, no evidence levelGroup of units of a r…conceptIdeal class group, concept, no evidence levelIdeal class groupconceptIdeal norm, concept, no evidence levelIdeal normconceptLattice (group), concept, no evidence levelLattice (group)conceptMinkowski's theorem, statement, no evidence levelMinkowski's theoremstatementOrder of a group, concept, no evidence levelOrder of a groupconceptQuadratic field, concept, no evidence levelQuadratic fieldconceptRegulator of a number field, concept, no evidence levelRegulator of a number…conceptRoot of unity, concept, no evidence levelRoot of unityconceptUnique factorisation domain, concept, no evidence levelUnique factorisation …conceptClass number problem, problem, no evidence levelClass number problemproblemRing, definition, no evidence levelRingdefinitionDirichlet's unit theorem, statement, no evidence levelDirichlet's unit theo…statementCommutative ring, definition, no evidence levelCommutative ringdefinitionSubring, definition, no evidence levelSubringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionClass number formula, statement, no evidence levelClass number formulastatementStark–Heegner theorem, statement, no evidence levelStark–Heegner theoremstatementMinkowski bound, statement, no evidence levelMinkowski boundstatementIntegral domain, definition, no evidence levelIntegral domaindefinitionFiniteness of the class number, statement, no evidence levelFiniteness of the cla…statementEuclidean domain, definition, no evidence levelEuclidean domaindefinitionGaussian integers, object, no evidence levelGaussian integersobjectNorm on the Gaussian integers, definition, no evidence levelNorm on the Gaussian …definitionThe Gaussian integers are a Euclidean domain, statement, no evidence levelThe Gaussian integers…statementClass number, definition, no evidence levelClass numberdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number field concept
  2. Binary quadratic form concept
  3. Class number concept
  4. Complex multiplication concept
  5. Dedekind domain concept
  6. Dedekind zeta function concept
  7. Dirichlet L-function concept
  8. Discriminant of a number field concept
  9. Free abelian group concept
  10. Group concept
  11. Group of units of a ring concept
  12. Ideal class group concept
  13. Ideal norm concept
  14. Lattice (group) concept
  15. Minkowski's theorem statement
  16. Order of a group concept
  17. Quadratic field concept
  18. Regulator of a number field concept
  19. Root of unity concept
  20. Unique factorisation domain concept
  21. Class number problem problem
  22. Ring definition
  23. Dirichlet's unit theorem statement
  24. Commutative ring definition
  25. Subring definition
  26. Zero divisor definition
  27. Class number formula statement
  28. Stark–Heegner theorem statement
  29. Minkowski bound statement
  30. Integral domain definition
  31. Finiteness of the class number statement
  32. Euclidean domain definition
  33. Gaussian integers object
  34. Norm on the Gaussian integers definition
  35. The Gaussian integers are a Euclidean domain statement
  36. Class number definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Ideal_class_group
About this page
Tag9BDJZGZ5Z5
URIm2://9BDJZGZ5Z5
Kinddefinition
WritingW0 stub
MSC11R29
DefinesClass number
UsesAlgebraic number field, Ideal class group, Order of a group, Finiteness of the class number, The Gaussian integers are a Euclidean domain, Dedekind domain, Stark–Heegner theorem
Changed2 November 2026

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