Class number

Be an algebraic number field. Then his class number is the order of the ( always finite) ideal class group of.

A prime number is called regular if, being a root of unity is.

Number theoretic meaning

If you want to solve an equation over a number field, then a possible strategy to solve the equation to the ideal group and the ideal class group. If 1 is the only solution to the ideal class group, then every ideal of a principal ideal. This number solves the original equation modulo units.

In order to solve the above equation, it is sufficient to know the structure of the Abelian group. In most cases, even the knowledge of the prime factorization of sufficient. ( for example, or. applicable)

For this reason, the determination of the ideal class number is one of the central tasks of number theory.

Example (special case of Fermat's last theorem )

Be an odd regular prime. Then the equation has no integer solutions.

Sketch of proof: The equation can be rewritten as. Going now to the ideals of above, we obtain, since the ideals on the left side are relatively prime, the equations. As the figure on the ideal class group of injective is obtained therefrom is the equations that can lead to a contradiction.

Properties

• Class number formula: For the class number:

• Be an extension, that is, and. Be the proportion of the class number. Then there is of independent natural numbers, so that for sufficiently large. ( See: Iwasawa Theory)
• Conjecture of Vandiver (not generally proved for verified ):

• For the following applies: for a
• Be. Then:

See also

• Relative class number