He took up Waring's problem in algebraic number fields and got interesting results.

By the going-down result above, no algebraic number field can be Euclidean.

Another notion needed to define algebraic number fields is vector spaces.

The notion of algebraic number field relies on the concept of a field.

Let K be an algebraic number field with R the ring of integers.

The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open.

The first step, however, is done in a different, more algorithmic efficiency way than the rational sieve, by utilizing algebraic number field.

See also algebraic number field.

Algebraic number fields and function fields over "'F"' are further global fields.

Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory.

Finite extensions of Q are also called algebraic number fields and are important in number theory.

There has been much work on Hilbert's tenth problem for the rings of integers of algebraic number fields.

LeVeque generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed algebraic number field.

The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871.

Landau prime ideal theorem for a generalization to prime ideals in algebraic number fields.

Obvious examples are the rings of integers of algebraic number fields as well as the rational numbers.

Hecke used a generalized theta series associated to an algebraic number field and a lattice in its ring of integers.

In general, this leads directly to the algebraic number field Q[r], which can be defined as the set of real numbers given by:

In number theory, a Parshin chain is a higher-dimensional analogue of a place of an algebraic number field.

Much later, the theory of Shimura provided another very explicit class field theory for a class of algebraic number fields.

The number R is called the regulator of the algebraic number field (it does not depend on the choice of generators u).

His early work was on algebraic number fields, how to decompose the ideal generated by a prime number into prime ideals.

As stated by Kaplansky, "The 11th Problem is simply this: classify quadratic forms over algebraic number fields."

Fields of algebraic numbers are also called algebraic number fields, or shortly number fields.

In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.