Algebraic Number Theory - James Milne

Algebraic Number Theory Milne Version March 18, 2017. 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 and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field Theory describes the abelian extensions of a Number field in terms of the arithmetic of the field. These notes are concerned with Algebraic Number Theory , and the sequel with class field Theory .

Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on.

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