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Algebraic Number Theory

Objectives

  • branch of number theory
  • use algebraic tools to study number-theoretic questions
  • number-theoretic questions are expressed in terms of algebra, eg. how do we decompose an ideal?

Unique factorization

  • irreducible element: x is irreducible if x = ab then a or b is unit.
  • prime: x is prime if x | ab then x | a or x | b
  • fact: all elements of a notherian ring can be decomposed into irreducible elements

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