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I'm requested to find the product of all quadratic irreducible polynomials of $\mathbf{Z}_{3}[x]$ . How can I find them ? brute force ? check that every polynomial has no roots ?

Or , if I take for example $f(x)=ax^2 + bx +c $ , find restrictions for a,b & c ?

I'd appreciate your help


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What does "power $2$" in the title mean? degree? Does the "multiplication" mean the product? The irreducible polynomials of degree $2$ in $\mathbb F_3[x]$ are minimal polynomials of elements of $\mathbb F_9 - \mathbb F_3 = \mathbb F_9^*- \mathbb F_3^*$, and hence their product is $(x^8-1)/(x^2-1) = x^6 + x^4 + x^2 + 1$. – Dilip Sarwate Mar 20 '12 at 11:45
Indeed you're correct , however I'd thank you if you could explain the meaning of $ F9−F3=F∗9−F∗3 $ and how you know those are the polynomials that you need to divide ... – ron Mar 20 '12 at 12:10
@ron, Dilip's suggestion is based on the fact that the product of minimal polynomials of $F_q$ is known to be $x^q-x$. The other key input is that the quadratic irreducible polynomials are exactly the minimal polynomials of $F_9\setminus F_3$. BTW, I will edit your question a bit. Please check. – Jyrki Lahtonen Mar 20 '12 at 12:41
Closely related to… if not a duplicate. – lhf Mar 20 '12 at 13:02
Why not "brute force"? It is a small problem. You probably mean monic quadratic irreducible. – André Nicolas Mar 20 '12 at 14:46

I think you are asking for all the polynomials of degree 2 irreducible over ${\bf Z}_3$. Let $f(x)=ax^2+bx+c$. You may assume $a=1$ (why?). You may assume $c\ne0$ (why?). That leaves you only six polynomials to check (why?).

Alternatively, you could argue that if it's not irreducible it must be a product of two linear factors. So once you get rid of $x^2,x(x-1),\dots,(x-2)^2$, the remaining (monic) polynomials must be irreducible.

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No, the query seeks only the product of all the irreducible quadratics, which doesn't require enumerating them all - see the comments to the question. – Bill Dubuque Mar 20 '12 at 13:42
@Bill, my answer went up before OP's comment and before Jyrki's edit. My answer still speaks to the question about "find(ing) restrictions for $a$, $b$, and $c$." – Gerry Myerson Mar 20 '12 at 22:32

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