---
# System prepended metadata

title: Modern Algebra

---

Modern Algebra
==============

Definitions and Theorems
------------------------

This document should become a quick-reference for the course, especially the part on Fields (Ch 21-23).

Throughout $G,H$ are **groups**, $R$ is a **ring**, $F,E,K$ are **fields**.

$H\le G$ means $H$ is a **subgroup** of $G$.

$H\triangleleft G$ means $H$ is a **normal** subgroup of $G$.

$[G:H]=|G/H|$ is the **index** of $H$ in $G$, i.e., the number of cosets of $H$.

$E\ge F$ means $E$ is a field extension of $F$.

$E>F$ means $E$ is a proper field extension of $F$.

$[E:F]=$ **degree** of the field extension  $=$ dim of $E$ as a vector space over $F$.

$R[x]$ is the **ring of polynomials** in $x$ with coefficients in $R$.

$F[x]$ is the ring of polynomials in $x$ with coefficients in $F$. Note that this is always an integral domain, in fact a UFD, hence a PID.

$F(x)$ is the **field of fractions** of $F[x]$.

$\alpha\in E$ is **algebraic over $F$** if $f(\alpha)=0$ for some $f(x)\in F[x]$.

$G(E/F)=$

A **primitive element** is a generator of the multiplicative group of the field (finite field GF(q))

The **fixed field** of a subgroup $G$ of Aut$(E)$ is $E_G=\{x\in E\mid \sigma(x)=x$ for all $\sigma\in G\}$.

Note the $E_G\subseteq E$. In fact, it is a subfield $E_G\le E$ (proof?)

$E$ is a **normal extension** of $F$ if ...

Normal is supposed to remind us of normal subgroups

Join the minecraft server main.mcparks.us if you miss Disneyland
bruh minecraft, less go
Is it modded or vanilla??
has to be modded
or i dont play (dont hate on vanilla ice cream, its good too)**gotta go neopolitan**best of both worlds hannah montana hahahahah
It isn't modded, but it looks modded. We really pushed the limits of what can be done in vanilla minecraft. You'd be quite surprised.
Basically, it is a 1:1 creation of a bunch of theme parks. Disneyland, DCA, Magic Kingdom, Animal Kingdom, EPCOT, Hollywood Studios, Universal Orlando, and Islands of Adventure. The rides are rideable, and there is music and shows and other fun things.




Hollywood 
Hello world Hot diggety dog I'm typing up a storm right now is anyone else typing??? I'm not sure but i guess I'll just keep typing GASP it's another **hahaah sabelle**
what color am i? like when i type what color is the cursor dude ... sad i see blue this is me idk who green is justine is blue lexy is green connor is pink chris is also green
Oh like a blue ore something
or youre green
one of the two
lexy is green nvm


**BOLD** ~~hey~~ ~Hey~ ^hey^ _hey_ $hey$ 
I am a !member!


Who is currently not a member? Yes, its a retorical question :)
hehe