Algebraic Number Theory by J.S. Milne

An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals in the ring of integers, the units, the extent to which the ring of integers fails to be have unique factorization, and so on. One important tool for this is “localization”, in which we complete the number field relative to a metric attached to a prime ideal of the number field. The com pleted field is called a local field — its arithmetic is much simpler than that of the number field, and sometimes we can answer questions by first solving them locally, that is, in the local fields.