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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 $F_p$ p is the amount of elements in the field. so for example for degree 4 in $F_2$ 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 $F_p$ p is the amount of elements in the field. so for example for degree 4 in $F_2$ 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 $\mathbb{F}_{p^d}$. Let $\alpha$ be a generator of its multiplicative group (the multiplicative group of any finite field is cyclic). Then $\mathbb{F}_{p^d}=\mathbb{F}_p(\alpha)$ and the minimal polynomial of $\alpha$ over $\mathbb{F}_p$, which is irreducible, has degree $\left[\mathbb{F}_p(\alpha):\mathbb{F}_p\right]=d$.

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