ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 06 Apr 2017 15:02:16 -0500second degree irreducible polynomial over extension fieldhttps://ask.sagemath.org/question/34197/second-degree-irreducible-polynomial-over-extension-field/ the given field is f_2^3 having third degree irreducible polynomial x^3+x+1 . if theta is a root of irreducible polynomial the field elements are {0,1,theta,theta^2,theta^3,theta^4,theta^5,theta^6}.
my question is i want second degree irreducible polynomial over f_{2^3}^2.Sat, 23 Jul 2016 05:51:30 -0500https://ask.sagemath.org/question/34197/second-degree-irreducible-polynomial-over-extension-field/Comment by Dima for <p>the given field is f_2^3 having third degree irreducible polynomial x^3+x+1 . if theta is a root of irreducible polynomial the field elements are {0,1,theta,theta^2,theta^3,theta^4,theta^5,theta^6}.</p>
<p>my question is i want second degree irreducible polynomial over f_{2^3}^2.</p>
https://ask.sagemath.org/question/34197/second-degree-irreducible-polynomial-over-extension-field/?comment=34260#post-id-34260I presume the question is about how to build a quadratic extension of GF(8) explicitly.Thu, 28 Jul 2016 03:08:22 -0500https://ask.sagemath.org/question/34197/second-degree-irreducible-polynomial-over-extension-field/?comment=34260#post-id-34260Comment by jaebond for <p>the given field is f_2^3 having third degree irreducible polynomial x^3+x+1 . if theta is a root of irreducible polynomial the field elements are {0,1,theta,theta^2,theta^3,theta^4,theta^5,theta^6}.</p>
<p>my question is i want second degree irreducible polynomial over f_{2^3}^2.</p>
https://ask.sagemath.org/question/34197/second-degree-irreducible-polynomial-over-extension-field/?comment=34228#post-id-34228I don't follow what you are asking for. Could you please clarify?Mon, 25 Jul 2016 22:22:10 -0500https://ask.sagemath.org/question/34197/second-degree-irreducible-polynomial-over-extension-field/?comment=34228#post-id-34228Answer by dan_fulea for <p>the given field is f_2^3 having third degree irreducible polynomial x^3+x+1 . if theta is a root of irreducible polynomial the field elements are {0,1,theta,theta^2,theta^3,theta^4,theta^5,theta^6}.</p>
<p>my question is i want second degree irreducible polynomial over f_{2^3}^2.</p>
https://ask.sagemath.org/question/34197/second-degree-irreducible-polynomial-over-extension-field/?answer=37202#post-id-37202A possible construction in the sens of the posting is as follows:
sage: R.<x> = PolynomialRing( GF(2) )
sage: F8.<t> = GF( 2**3, modulus=x^3+x+1 )
sage: F8
Finite Field in t of size 2^3
sage: RF8 = PolynomialRing( F8 )
We have so far $\mathbb{F}_8$, realized so that $t$ is a generator of the multiplicative group
$\mathbb{F}_8^\times$ with $7=(8-1)$ elements. Starting from it, we can extend, by explicitly introducing also a root of order nine (but not $3$) of unity. So that we have one of order $63=(8-1)(8+1)=64-1$.
So we extend:
sage: RF8.<y> = PolynomialRing( F8 )
sage: factor( y^9-1 )
(y + 1) * (y^2 + y + 1) * (y^2 + t*y + 1) * (y^2 + t^2*y + 1) * (y^2 + (t^2 + t)*y + 1)
sage: F64.<u> = F8.extension( y^2 + t*y + 1 )
sage: F64.is_field()
True
sage: F64.order()
64
sage: u^3, u^9
((t^2 + 1)*u + t, 1)
sage: g = u*t
sage: g^7, g^9, g^63
(t*u + t^2 + 1, t^2, 1)
So $u$ has multiplicative order $9$, and $g=ut$ correpsondingly $63$. The modulus for the constucted field, so that $g$ is a root of it is:
sage: for f,_ in factor( x^63-1 ):
if f(g) == 0:
print f
....:
x^6 + x^4 + x^3 + x + 1
sage: g^6 + g^4 + g^3 + g + 1
0
Thu, 06 Apr 2017 15:02:16 -0500https://ask.sagemath.org/question/34197/second-degree-irreducible-polynomial-over-extension-field/?answer=37202#post-id-37202