- Find all monic irreducible polynomials of degree 2 over Z3.

x^2 + 1, x^2 + x + 2,x^2 + 2x + 2, this is the answer from the book. But I have no idea what it actually means. Can someone explain?

-Wooi-Chen Ng

Monic means the leading coefficient is 1. Degree two means the highest power is 2. And irreducible means it doesn't factor interestingly. So each polynomial has x^2 as the leading term. Then we just add all the possible combinations of linear polynomials in Z3 and see which do not factor.

-Allen

Since the Reducibility Test for Degrees 2 and 3 (Theorem 17.1) is an if-and-only-if statement, it can also be used instead of trying to factor to determine which ones are the irreducible ones.

-Josh

There are 6 different polynomials that need to be checked since we are in Z_3 = {0,1,2}. I did what Josh did and used the Reducibility Test for Degrees 3 and 3 to determine which ones are the irreducible ones. The only ones that give roots [f(x)=0] are the answers shown above. The only way to get the roots though is to be in Z_3. So for example f(x) = x^2 + 2 when I put in 1 I get 3 but in Z_3 equals 0 and thus why it is irreducible.

--Robertsr 09:57, 13 November 2008 (UTC)