Relative class number

The relative class number is a mathematical term from the field of algebraic number theory .

Let K be an Abelian number field , d. H. a finite, Galois expansion of the body with an Abelian Galois group . (According to Kronecker-Weber's theorem , K is a subfield of a circle dividing field .) Let K be embedded in the complex numbers and be the real subfield. Let be the class number of K and that of . Then integer and is called the relative class number of K . ${\ displaystyle K / Q}$ ${\ displaystyle K ^ {+}: = K \ cap \ mathbb {R}}$ ${\ displaystyle h}$ ${\ displaystyle h ^ {+}}$ ${\ displaystyle K ^ {+}}$ ${\ displaystyle h ^ {-}: = {\ frac {h} {h ^ {+}}}}$

More generally, this construction is possible for CM bodies , i.e. H. imaginary square extensions of totally real number fields . A number field is totally real if the image of each embedded in is included. "CM" stands for complex multiplication and indicates the connection with Abelian varieties with complex multiplication. ${\ displaystyle K}$ ${\ displaystyle K \ to \ mathbb {C}}$ ${\ displaystyle \ mathbb {R}}$

The class numbers of the circle division bodies are important for the historical proof approaches of the great Fermat's theorem (see Great Fermat's theorem # All regular prime numbers as well as regular prime numbers ). The relative class number appears in the intermediate step

{\ displaystyle p \ mid h_ {p} \ iff p \ mid h_ {p} ^ {-} \ iff p \ mid B_ {j} \ {\ text {for a}} \ j = 2,4, \ ldots , p-3}

on (cf. Bernoulli numbers ).

literature

Lawrence C. Washington : Introduction to cyclotomic fields (= Graduate Texts in Mathematics. Vol. 83). Springer, New York NY et al. 1982, ISBN 0-387-90622-3 .