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

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November 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)).

Please help, im lost on this one.
November 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)).

Please help, im lost on this one.

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
November 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?
January 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
January 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 $\mathbb{F}_3[x]/<x^2+1>$ 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
January 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
January 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
January 27th 2011, 06:42 AM
Mike12

thanks for replying , yes I meant finding the all factors
please , help me
January 27th 2011, 06:57 AM
HallsofIvy

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.