817 - Algebra - Revision history

Kfagerstrom: /* Integral Domains */

2023-01-17T20:44:50Z

Integral Domains

2023-01-17T20:40:44Z

Prime Ideals

2023-01-17T20:38:35Z

Semi-Direct Product

2023-01-17T20:36:13Z

Sylow Theory

2023-01-17T20:01:20Z

Named Rings

2023-01-16T15:41:38Z

Sylow Theory

2023-01-16T15:03:53Z

Sylow Theory

2023-01-16T14:58:42Z

Groups

2022-12-07T23:05:21Z

2022-12-07T16:22:26Z

Prime Ideals

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 ===Named Rings===
-====Integral Domains====
+====[[Integral Domains]]====
 An ''Integral Domain'', often just called domain, is a commutative ring <math>R </math>, with <math>1\neq 0 </math> and has no zero divisors.
 ==== Euclidian Domains (EDs) ====

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-====Prime Ideals ====
+====[[Prime Ideals]] ====
 A ''prime ideal'' of a ring <math>R </math> is proper ideal <math>P </math> such that whenever <math>xy\in P </math> for <math>x,y \in R </math>, <math>x\in P </math> or <math>y \in P </math>. </br>
 An ideal <math>P </math> is prime if and only if <math> R\backslash P</math> is closed under multiplication.

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 We set <math> \text{Syl}_p(G)</math> to be the collection of all Sylow p-subgroups of <math>G </math> and <math>n_p = | \text{Syl}_p(G)| </math> to be the number of Sylow p-subgroups.
-====Semi-Direct Product ====
+====[[Semi-Direct Product]] ====
-:''see also'' [[Semi-Direct Products]]
 ====Quotient Groups====
 ====Isomorphism Theorems ====

@@ Line 5: / Line 5: @@
 ===Topics in Group Theory ===
-====Sylow Theory====
+====[[Sylow Theory]]====
-:''see also'' [[Sylow Theory]]
 Let G be a finite group and p a prime. Write the order of <math> G</math> as

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 ===Named Rings===
 ==== 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) ====

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 Let G be a finite group and p a prime. Write the order of <math> G</math> as
-<math> |G| = p^em </math> where <math>p \not | m</math>. A Sylow p-subgroup of <math> G</math> is a subgroup <math> H \leq G </math>such that <math> |H| = p^e</math>. That is, a Sylow p-subgroup of <math>G </math> is a subgroup whose order is the highest conceivable power of <math>p </math> according to Lagrange’s Theorem.
+<math> |G| = p^em </math> where <math>p \not| m</math>. A Sylow p-subgroup of <math> G</math> is a subgroup <math> H \leq G </math>such that <math> |H| = p^e</math>. That is, a Sylow p-subgroup of <math>G </math> is a subgroup whose order is the highest conceivable power of <math>p </math> according to Lagrange’s Theorem.
 We set <math> \text{Syl}_p(G)</math> to be the collection of all Sylow p-subgroups of <math>G </math> and <math>n_p = | \text{Syl}_p(G)| </math> to be the number of Sylow p-subgroups.