# Algebra help: Find a primitive root β of F3[x]/(x^2 + 1)...etc

• Nov 24th 2010, 01:41 PM
habsfan31
Algebra 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)).

• Nov 24th 2010, 06:34 PM
tonio
Quote:

Originally Posted by habsfan31
(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)).

If you meant $\mathbb{F}_3[x]=$ the ring of pol's over the field with three elements, then $\mathbb{F}_3[x]/(x^2+1)$ is a

field with 9 elements (why?) , so $\beta$ is a generator of the multiplicative group of this field. What have you tried/done?

Tonio
• Nov 24th 2010, 06:40 PM
habsfan31
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 PM
Mike12
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 PM
tonio
Quote:

Originally Posted by Mike12
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

Suppose $w$ is an element in some extension of $\mathbb{F}_3$ s.t. $w^2+1=0\Longleftrightarrow w^2=-1$ , so every

element in $\mathbb{F}_3[x]/$ is of the form $aw+b$ , with $a,b\in\mathbb{F}_3$ and

sum and multiplication modulo 3, taking into account that $w^2=-1$.

Thus, for example, $w\cdot (2w+1)=2w^2+w=(-2)+w=1+w$ , etc.

Tonio
• Jan 26th 2011, 07:07 PM
Mike12
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 AM
tonio
Quote:

Originally Posted by Mike12
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

For any natural number $n$ , $x^n-1=(x-1)(x^{n-1}+x^{n-2}+\ldots+x+1)$

By "multiple factor" I understand that you meant that the polynomial factors...right?!

Tonio
• Jan 27th 2011, 06:42 AM
Mike12
thanks for replying , yes I meant finding the all factors
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 " $F_2$" is?
In $F_2$, 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 $(x- 1)(x^4+ x^2+ 1)$ Now, if $g(x)= x^4+ x^2+ 1$, 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 $F_2(x)$ are of the form $x^2+ ax+ b$ with a and b either 0 or 1. That is, there are only four possible quadratics: $x^2$, $x^2+ 1$, $x^2+ x$, and $x^2+ x+ 1$. You could try dividing $x^4+ x^2+ 1$ by those (being careful to use " $F_2$" arithmetic) to see if it factors and, if so, what those factors are.