# 0x01 Algebra

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

Contents Index

## 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
• 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

## Module

Definitions:

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