Galois Theory Notes

Date: 2023/11/10
Last Updated: 2024-03-07T20:23:34.701Z
Categories: Math
Tags: Galois Theory, Math, Group Theory, Field Theory, Notes
Read Time: 2 minutes

Contents

Intro

This is my personal note for Galois Theory.

This is a work in progress. When I have time, I will add more content to it.

Concepts of Ring

Ring is a set equipped with two binary operations, usually referred to as addition $+$ and multiplication $·$, that satisfy the following conditions:
  1. The set is an abelian group under the operation ++ with an identity element usually denoted as 0.

  2. The operation · is associative, meaning that for all a,b,ca, b, c in the set, (ab)c(a·b)·c equals a(bc)a·(b·c).

  3. The operation · is distributive over the operation ++, meaning that for all a,b,ca, b, c in the set, a(b+c)a·(b+c) equals ab+aca·b + a·c and (a+b)c(a+b)·c equals ac+bca·c + b·c.

  4. The operation · has an identity element, usually denoted as 11.

  5. The operation · is closed, meaning that for all a,ba, b in the set, aba·b is also in the set.

  6. A ring is called a commutative ring if the operation · is commutative, meaning that for all a,ba, b in the set, aba·b equals bab·a.

Field is a ring in which every non-zero element is invertible.
Subring is a subset of a ring that is a ring with the restriction of the ring operations.
Subfield is a subset of a field that is a field with the restriction of the field operations.
Ring Homomorphism is a function between two rings that preserves the operations of the rings.

A ring homomorphism is called an isomorphism if it is bijective.

Ideal is a subset of a ring that is closed under addition, negation, and multiplication by any element in the ring.
Principal Ideal is an ideal that is generated by a single element.
Integral Domain is a commutative ring with identity in which the product of any two non-zero elements is non-zero.
Quotient Ring is a ring constructed from a ring $R$ and an ideal $I$ of $R$. Where the elements of the quotient ring are the cosets of $I$ in $R$.

Definitions


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