My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
November 30th, 2017, 03:57 AM   #1
Member
 
Joined: Jan 2015
From: usa

Posts: 80
Thanks: 0

subgroups of a Galois group

Please help me to answer the following problem:

Let $f(x)=x^6+3\in\mathbb{Q}[x]$.
Assume the following facts:

1) $f$ is irreducible in $\mathbb{Q}[x]$. Let $\alpha$ be a root of $f(x)$ and $L=\mathbb{Q}(\alpha)$.

2) $L$ is a splitting field for $f(x)$ over $\mathbb{Q}$.

3) $Gal(L/\mathbb{Q})$ is isomorphic to a copy of $S_3$ inside $S_6$.

>For each non-trivial proper subgroup $H<Gal(L/\mathbb{Q})$ determine $Fix(H)$, Express each answer in the form $\mathbb{Q}(\beta)$ with $\beta$ given explicitly in terms of $\alpha$ and also give the minimal polynomial of $\beta$ over $\mathbb{Q}$

Last edited by skipjack; December 19th, 2017 at 08:25 PM.
mona123 is online now  
 
December 19th, 2017, 05:22 AM   #2
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 2,918
Thanks: 783

Have you determined the possible values of $\displaystyle \alpha$. That is, have you determined all 6 roots of $\displaystyle x^6+ 3= 0$?

Last edited by skipjack; December 19th, 2017 at 08:24 PM.
Country Boy is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
galois, group, subgroups



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Fixed fields and Galois subgroups. Artus Abstract Algebra 0 May 9th, 2013 06:58 PM
Galois group goodfeeling Abstract Algebra 2 August 29th, 2012 12:36 PM
Galois group of x^5-5x+12 mathbalarka Abstract Algebra 1 July 11th, 2012 10:34 AM
Normal Subgroups & Galois Theory... TTB3 Abstract Algebra 6 June 27th, 2009 07:57 AM
Galois Group, FTG diondex223 Abstract Algebra 1 March 17th, 2009 09:19 AM





Copyright © 2018 My Math Forum. All rights reserved.