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
September 28th, 2012, 09:49 AM   #1
Senior Member
 
Joined: Sep 2010
From: Germany

Posts: 153
Thanks: 0

generating a maximal ideal, is this correct?

Could someone verify If my way of thinking here is correct?
let and we know that maximal ideals of the ring has the form where p is prime and .

1) we choose an irreducible polynomial and then we lift  any way we want to polynomial

say I choose  and then the pair is a maximal ideal of and is a finite algebraic extension field.

some other example
 and then I lift the polynomial mod 2 to or
to so I can get 2 maximal ideals


Is this right?
Thanks
rayman is offline  
 
October 4th, 2012, 11:57 AM   #2
Senior Member
 
Joined: Sep 2008

Posts: 150
Thanks: 5

Re: generating a maximal ideal, is this correct?

Unfortunately, it is not easy to see what exacly you are trying to show. Most of what you say is correct up to minor typos.

Quote:
Originally Posted by rayman
Could someone verify If my way of thinking here is correct?
let and we know that maximal ideals of the ring has the form where p is prime and .
It is correct that every maximal ideal of is of that form, yes.
Quote:
1) we choose an irreducible polynomial and then we lift any way we want to polynomial
yes, this construction will give exacly the maximal ideals.
Quote:
say I choose and then the pair is a maximal ideal of and is a finite algebraic extension field.
I suppose all the 2's in the index should be 3's, then it is true, that you will get a finite extension of , but it will be a proper extension, i.e. the "="-sign in is wrong.
Quote:
some other example
and then I lift the polynomial mod 2 to or
to so I can get 2 maximal ideals


Is this right?
Thanks
Well it is true, however it is worth noting, that the ideals you get by chosing diffrent lifts of the polynomial will allways be the same. In particular m_1=m_2. To get different ideals you can take either different p's or take different irreducible polynomials in .
Peter is offline  
October 4th, 2012, 11:06 PM   #3
Senior Member
 
Joined: Sep 2010
From: Germany

Posts: 153
Thanks: 0

Re: generating a maximal ideal, is this correct?

thank you for your reply
I just wanted to find out some systematical way of producing these ideals.
So these polynomials can be of any degree as long as they are irreducible right?
rayman is offline  
October 6th, 2012, 09:17 AM   #4
Senior Member
 
Joined: Sep 2008

Posts: 150
Thanks: 5

Re: generating a maximal ideal, is this correct?

correct. As long, as by irriduceable you imply ireducible as a polynomial over . The degree of the polynomial (mod p) will then be the degree of the (finite algebraic) field extension .
Peter is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
correct, generating, ideal, maximal



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Maximal ideal limes5 Abstract Algebra 7 January 6th, 2014 03:43 PM
maximal ideal cummings123 Abstract Algebra 1 February 27th, 2013 05:06 AM
maximal ideal of the ring Q -trivial question?? rayman Abstract Algebra 2 November 21st, 2012 03:16 AM
ideal and maximal silvi Abstract Algebra 2 January 20th, 2010 03:05 AM
Maximal Ideal julien Abstract Algebra 1 November 19th, 2006 05:56 PM





Copyright © 2019 My Math Forum. All rights reserved.