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November 7th, 2012, 10:23 AM   #1
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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)
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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.
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February 1st, 2013, 02:03 PM   #3
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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 .
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