The reason for this distinction relates to general algebraic number theory.

A separate inspiration for F came from algebraic number theory.

Class field theory is the key part and the heart of algebraic number theory.

Almost any book on modern algebraic number theory, such as:

For more information about that: read algebraic number theory.

He is known for his work in the areas of automorphic forms, and algebraic number theory.

In algebraic number theory one defines also norms for ideals.

This is one of the main results of classical algebraic number theory.

Another example, playing a key role in algebraic number theory, is the field Q of p-adic numbers.

Other books of his covered projective geometry and algebraic number theory.

The totally real number fields play a significant special role in algebraic number theory.

Important examples of locally profinite groups come from algebraic number theory.

His other interests included algebraic number theory, mathematical economics and geometry of numbers.

The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory.

Their detailed properties are studied in algebraic number theory.

Thirdly, groups were (first implicitly and later explicitly) used in algebraic number theory.

The Langlands Program is a branch of algebraic number theory.

This intuition also serves to define ramification in algebraic number theory.

This spirit is adopted in algebraic number theory.

He has also written several widely-used textbooks in computational and algebraic number theory.

The recognition of this failure is one of the earliest developments in algebraic number theory.

Pisano periods can be analyzed using algebraic number theory.

They are used in algebraic number theory and algebraic topology.

In algebraic number theory, tensor products of fields are (implicitly, often) a basic tool.

Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory.