(a) Find a primitive root β of F3[x]/(x^2 + 1).

(b) Find the minimal polynomial p(x) of β in F3[x].

(c) Show that F3[x]/(x^2 + 1) is isomorphic to F3[x]/(p(x)).

Please help, im lost on this one.

- Nov 24th 2010, 01:41 PMhabsfan31Algebra help: Find a primitive root β of F3[x]/(x^2 + 1)...etc
(a) Find a primitive root β of F3[x]/(x^2 + 1).

(b) Find the minimal polynomial p(x) of β in F3[x].

(c) Show that F3[x]/(x^2 + 1) is isomorphic to F3[x]/(p(x)).

Please help, im lost on this one. - Nov 24th 2010, 06:34 PMtonio
- Nov 24th 2010, 06:40 PMhabsfan31
I havent really tried anything, im pretty lost when it comes to this topic and the book is really confusing. I already handed in the assignment with this question blank but would still like to know how to solve it. Can you possibly show me how its done step by step?

- Jan 26th 2011, 05:18 PMMike12
Can you tell me please how does the elements in F_3[x]/ (x^2+1) look like in order to be able to construct the addition and multiplication tables

- Jan 26th 2011, 06:28 PMtonio
- Jan 26th 2011, 07:07 PMMike12
can you tell me please how to prove that f(x)=x^5+x^4+x^3+x^2+x+1 belongs to F_2[x] has a multiple factor. and how can I find the all factors of f(x) and their multiplicities

- Jan 27th 2011, 03:16 AMtonio
- Jan 27th 2011, 06:42 AMMike12
thanks for replying , yes I meant finding the all factors

please , help me - Jan 27th 2011, 06:57 AMHallsofIvy
This concerns me. You seem to know nothing about what, if you know the basic definitions, is a fairly simple problem just involving "arithmetic". Are you clear on what " "

**is**?

In , the only "numbers" are 0 and 1: 0+ 0= 1, 0+1= 1+0= 1, 1+1= 0; 0*0= 0*1= 1*0= 0, 1*1= 1. If x= 0, then f(0)=0+ 0+ 0+ 0+ 0+ 1 but if x= 1, then f(1)= 1+1+ 1+ 1+ 1+ 1= 0 just because there are an even number of terms. That's how we know x-1 is a factor and, by synthetic division, that it is equal to Now, if , then g(0)= 0+0+ 1= 1 and g(1)= 1+ 1+ 1= 1 so g has no linear factors and so no third degree factors. The only possible way to factor g would be as two quadratic factors and the only possible "quadratic" polynomials in are of the form with a and b either 0 or 1. That is, there are only four possible quadratics: , , , and . You could try dividing by those (being careful to use " " arithmetic) to see if it factors and, if so, what those factors are.