How to Determine Galois Groups and Subfields?

  • Thread starter Thread starter minibear
  • Start date Start date
  • Tags Tags
    Group
minibear
Messages
6
Reaction score
0
I have trouble in determining Galois group.
Can anyone help me with the following question:

how to determine the Galois group of (x^2-2)(x^2-3)(x^2-5), determine all the subfields of the splitting field of this polynomial?

how to determine the elements of the galois group of x^p-2 for p is prime.

Thanks a lot!
 
Physics news on Phys.org
The set of solutions of xp= a are of the form |a|^{1/p}\omega^i with i ranging from 0 to p-1, where \omega is the "principal pth root of unity". The Galois group is the permutation group of that set: the permutation group on p objects.
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Proving properties of polynomial in K[x]
Replies
8
Views
2K
I Generators of Galois group of ## X^n - \theta ##
Replies
3
Views
2K
I Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
Replies
0
Views
2K
A Galois theorem in general algebraic extensions
Replies
6
Views
2K
I Dfns group action & group, written in terms of the other
Replies
26
Views
377
MHB Automorphisms, Galois Groups & Splitting Fields
Replies
3
Views
2K
I Galois Theory - Fixed Field of F and Definition of Aut(K/F)
Replies
2
Views
3K
A Structure of modulo-integer matrix group GL(r,Z(n))?
Replies
17
Views
6K
When do roots of a polynomial form a group?
Replies
2
Views
2K
MHB Galois Theory - Fixed Field of F and Definition of Aut(K/F) ....
Replies
2
Views
3K

Hot Threads

Recent Insights

Back
Top