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 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
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Re: finite fields

Quote:
 Originally Posted by gelatine1 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 f(x) = x^4+x^3+x^2+1 (note highest coefficient is 1 because there are only 2 elements in the field: 0 and 1)
f(1) = 0, so this polynomial has x-1 as a factor, and is not irreducible.

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 Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post gelatine1 Number Theory 0 November 7th, 2012 07:44 AM lreps Abstract Algebra 1 April 12th, 2011 01:07 PM lreps Abstract Algebra 0 April 10th, 2011 12:38 PM Mathworm Abstract Algebra 1 March 5th, 2009 02:20 PM lreps Number Theory 0 December 31st, 1969 04:00 PM

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