Introduced the concept of Algebraic NumbersĀ and proved some of their properties: Countability (Cantor), finite degree field extension, algebraically closed in fields or rings

Proved the existence of Transcendental Numbers (Liouville).

Discussed some properties of minimal polynomials

Unique monic minimal polynomial having a root as algebraic number

Irreducible implies distinct roots in Complex field

If a is algebraic, the ring of Q joined by a is equivalent to the field of Q joined by a

Theorem: If K is a subset of L, and is a finite extension of field with characteristic 0, then L is a field extension of K.

Introduced the concept of Algebraic Integer, and that any monic polynomial with coefficient being integer will be an algebraic integer

Introduced the concept of Module, considered rings of algebraic integers as modules, connection with linear algebra.

Tool: if d = 2,3 mod 4, then Z joined by root d will be normal, while d = 1 mod 4, Z joined by root d will be (1+root(d))/2.

Euclidean Algorithm and it’s expansion into other domains, like polynomials and imaginary fields. So as the Bezout algorithm.

Reviewed the Fundamental Theorem of Arithmetic.

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.

Discussed why the Euclidean Algorithm fails in some field extensions.

Solved equations in the third or higher power in other domains.