Algebraic Number Theory Outline 1

  1. Euclidean Algorithm and it’s expansion into other domains, like polynomials and imaginary fields. So as the Bezout algorithm.
  2. Reviewed the Fundamental Theorem of Arithmetic.
  3. Theorem: Every prime number p = 1 mod 4 can be written as the sum of two squares. (Proof intuition, Wilson’s Theorem) More generalized Theorem: A natural number could be written as the sum of two squares iff its 3 (mod 4) prime factorizations have even power.
  4. Discussed why the Euclidean Algorithm fails in some field extensions.
  5. Solved equations in the third or higher power in other domains.