Re: irreducible polynomial in Z_7[t] roots of which are primitive in GF(49)
- From: anonymous.rubbertube@xxxxxxxxx
- Date: Fri, 14 Nov 2008 13:28:14 -0800 (PST)
On Nov 14, 4:06 pm, Derek Holt <ma...@xxxxxxxxxxxxx> wrote:
On 14 Nov, 13:22, anonymous.rubbert...@xxxxxxxxx wrote:
On Nov 14, 7:57 am, Kenneth Bull <kenneth.b...@xxxxxxxxx> wrote:
How to find an irreducible polynomial in Z_7[t] roots of which are
primitive in GF(49) >
Primitives in GF(p^n) are primitive (p^n - 1)th roots of unity; so to
get a polynomial in Z_p[t] whose roots are primitives in GF(p^n), try
a polynomial whose splitting field is the degree n unramified
extension of Z_p. In your case, I think the 48th (48 = 7^2 - 1)
cyclotomic polynomial does it.
The 48th cyclotomic polynomial is x^16 - x^8 + 1, and its
factorization over Z_7 is
<x^2 + x + 3>*<x^2 + 2*x + 3>*<x^2 + 2*x + 5>*<x^2 + 3*x + 5>*
<x^2 + 4*x + 5>*<x^2 + 5*x + 3>*<x^2 + 5*x + 5>*<x^2 + 6*x + 3>.
Derek Holt
Then I suppose those quadratic factors are the irreducible polynomials
that the OP is looking for.
.
- Follow-Ups:
- References:
- irreducible polynomial in Z_7[t] roots of which are primitive in GF(49)
- From: Kenneth Bull
- Re: irreducible polynomial in Z_7[t] roots of which are primitive in GF(49)
- From: anonymous . rubbertube
- Re: irreducible polynomial in Z_7[t] roots of which are primitive in GF(49)
- From: Derek Holt
- irreducible polynomial in Z_7[t] roots of which are primitive in GF(49)
- Prev by Date: Re: Theorem on Natural Numbers
- Next by Date: Re: Is there really that big of a change in high school geometry teaching in the past 15 years?
- Previous by thread: Re: irreducible polynomial in Z_7[t] roots of which are primitive in GF(49)
- Next by thread: Re: irreducible polynomial in Z_7[t] roots of which are primitive in GF(49)
- Index(es):
Relevant Pages
|
