In mathematics, algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. 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 behaviour 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 behaviour of the usual integers.

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