November 7th, 2012, 10:23 AM  #1 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11  finite fields
How do i find irreducible polynomials of degree d and with numbers of a field p is the amount of elements in the field. so for example for degree 4 in i got x^4+x^3+x^2+1 (note highest coefficient is 1 because there are only 2 elements in the field: 0 and 1) 
January 13th, 2013, 08:58 PM  #2  
Newbie Joined: Jan 2013 Posts: 6 Thanks: 0  Re: finite fields Quote:
One way to find irreducible polynomials is to look at F_4. It has two elements not in F_2. Calculate their minimal polynomial over F_2. You can keep going on to F_8, etc. The question about finding minimal polynomials of specified degree, though, I don't know.  
February 1st, 2013, 02:03 PM  #3 
Member Joined: Jan 2013 Posts: 93 Thanks: 0  Re: Finite fields
I think you can prove it as follows. Consider the finite field . Let be a generator of its multiplicative group (the multiplicative group of any finite field is cyclic). Then and the minimal polynomial of over , which is irreducible, has degree . 

Tags 
fields, finite 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
finite fields  gelatine1  Number Theory  0  November 7th, 2012 07:44 AM 
Finite fields  lreps  Abstract Algebra  1  April 12th, 2011 01:07 PM 
Finite fields  lreps  Abstract Algebra  0  April 10th, 2011 12:38 PM 
Pseudofinite fields  Mathworm  Abstract Algebra  1  March 5th, 2009 02:20 PM 
Finite fields  lreps  Number Theory  0  December 31st, 1969 04:00 PM 