0x010 Abstract Algebra

Group

Definition (group) A group is an ordered pair $(G, *)$ where $G$ is a set and $*$ is a binary operation on $G$ satisfying following axioms

  • associative: $(a*b)*c = a*(b*c)$
  • identity: $(\exists e \in G) (\forall a \in G) a*e = e*a = a$
  • inverse: $(\forall a \in G) (\exists a^{-1} \in G) a*a^{-1} = a^{-1}*a = e$

Definition (Abelian) The group is called abelian if $a*b = b*a$

Ring

Modules

Field

Galois

Reference

[1] Dummit, David Steven, and Richard M. Foote. Abstract algebra. Vol. 3. Hoboken: Wiley, 2004.