0x01 Algebra

This page will mainly contain my summary of GTM 73 Algebra and a couple of other course materials.

Group

structure which add and sub are defined

Definitions:

  • semi-group: (R, +) with associativity
  • monoid: (R, +) with associativity and identity
  • group: (R, +) where R is a set, and + is an operation
    • associativity, identity, inverse
  • abelian: group with commutativity

Theorems:

  • Lagrange: The order of every subgroup H of finite group G divides the order of G
  • Sylow: For every prime factor p with multiplicity n of the order of a finite group G, there exists a subgroup of G of order $p^n$.

Ring

structure which add, sub, mul are defined

Definitions:

  • ring: (R, +, x) where R is a set, + and x are operations
    • addition is abelian
    • multiplication is monoid
    • multiplication is distributive over addition
  • semi-ring: ring without additive inverse
  • ideal: a subring which is closed under mulplication between the subring elem and ring elem

Field

Definitions

  • Field: structure which add, sub, mul, div are defined

Finite Field

Definition:

  • Finite Field (Galois Field): field whose number element is finite
  • Prime Field: finite field with prime order $GF(p)$

Properties

  • Every finite field is isomorphic to $F_{p^n}$ where p is prime and n is positive

Galois

Linear Algebra

Module

Definitions:

  • module: module (abelian group) over a ring is a generalization of the notion of vector space over a field

Category