Algebraic number theory

Algebraic number theory is a branch of number theory, which in turn is a branch of mathematics.

The algebraic number theory goes beyond the whole and rational numbers addition and considered algebraic number fields, which are finite extensions of the rational numbers. Elements of number fields are roots of polynomials with rational coefficients. This number fields contain the integers analogous subsets that wholeness rings. They behave in many respects like the ring of integers, but some properties take on a slightly different form. For example, there is generally no unique decomposition into prime numbers more, but only in prime ideals.

The algebraic number theory still employed with the study of algebraic function fields over finite fields, the theory of which is largely analogous to the theory of number fields. Algebraic number and function fields are grouped under the name of " global body."

Often it turns out to be fruitful questions " locally ", ie, to consider separately for each prime spot (local -global principle). This process results in the case of the integers to the p- adic numbers, general to local bodies.

Further terms

• Dedekind ring
• Global body
• Local body
• Class field theory