Algebra help - primitive roots and minimal polynomials

[cooljosh2k2]

Homework Statement

(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)).

The Attempt at a Solution

I am completely lost on this one

 

[micromass]

So for the first one, you need to find a primitive root in $$\mathbb{F}_3[x]/(x^2+1)$$
Can you tell us what a primitive root is?
Can you find any root (not just primitive) in our field? To find this, take an arbitrary element a+bx in our field. Can you find a, b and n such that $$(a+bx)^n=1$$?

 

[cooljosh2k2]

From what i understand, a primitive root is a value that when taking to the power of the order of the polynomial, you will get 1 (mod 3 for this example) I don't know if this is right, but i get x^2+1 as a primitive root. Does that make sense. The order of the polynomial is 2, so (x^2+1)^2 = 1

 

[micromass]

Wow, I have an entirely different definition of primitive root...
I have this: Let F be a field, then a nth root is an element x in F such that $$x^n=1$$ for some $$n>0$$. The nth root x is called primitive if there is no k<n such that $$x^k=1$$.

Does this make any sense to you? Or do you understand something completely different in the term root?

 

[cooljosh2k2]

That makes sense, but from what i thought i understood, the n is usually the order (or degree) of the polynomial. But i might be wrong.

 

[micromass]

I don't really understand what polynomials have to do with this. The only thing I can image is that a nth root of unity is a root of the polynomial $$X^n-1$$.

Can you give me the exact definition in your course? I really want to know if we're talking about thesame thing here...