Algebraic number theory

Algebraic number theory is a major branch of number theory that studies algebraic structures related to algebraic integers.

Description

This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behavior of ideals, and field extensions.

In this setting, the familiar features of the integers—such as unique factorization—need not hold. The virtue of the primary machinery employed—Galois theory, group cohomology, group representations, and L-functions—is that it allows one to deal with new phenomena and yet partially recover the behavior of the usual integers.

See also

• Adele ring
• Arithmetic algebraic geometry
• Class field theory
• Ideal (ring theory)
• Ideal class group
• Iwasawa theory
• Kummer theory
• Langlands program
• Tamagawa number

External links

• Algebraic number theory @ Wikipedia