Re: irreducible polynomial in Z_7[t] roots of which are primitive in GF(49)

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
.

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