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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 
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:
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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? 
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 .


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