Difference between revisions of "817 - Algebra"
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==== Euclidian Domains (EDs) ==== | ==== Euclidian Domains (EDs) ==== | ||
| + | A ''Euclidean Domain'' (ED) is a domain <math>R </math> together with a function <math>N: R \to \mathbb{Z}_{\geq 0} </math> such that <math>N(0)=0 </math> and the following property holds: for any two elements <math> a,b \in R </math> with <math>b\neq 0 </math>, there are elements <math> q,r \in R</math> such that <math> a=qb+r </math> and either <math>r=0 </math> or <math>N(r)<N(b) </math> | ||
| + | |||
====Principle Ideal Domains (PIDs) ==== | ====Principle Ideal Domains (PIDs) ==== | ||
====Unique Factorization Domains (UFDs) ==== | ====Unique Factorization Domains (UFDs) ==== | ||
Revision as of 15:48, 7 December 2022
Contents
Groups
Theorems
Topics in Group Theory
Sylow Theory
Semi-Direct Product
Quotient Groups
Isomorphism Theorems
Rings
Definition: A ring is a set \(R\) with two binary operation \( +\) and \(\cdot\) satisfying:
- \( (R,+)\) is an abelian group (with identity 0)
- \( (R,\cdot) \) is a semigroup
- \(\cdot \) is distributive over \( + \) (on both sides)
Named Rings
Euclidian Domains (EDs)
A Euclidean Domain (ED) is a domain \(R \) together with a function \(N: R \to \mathbb{Z}_{\geq 0} \) such that \(N(0)=0 \) and the following property holds: for any two elements \( a,b \in R \) with \(b\neq 0 \), there are elements \( q,r \in R\) such that \( a=qb+r \) and either \(r=0 \) or \(N(r)<N(b) \)
