The problem is:
Count the number of irreducible monic polynomials of degree 2 in (Z/13Z)[X]. Prove your answer.
What would be the best way to do this problem, or any tips?
First, you're looking for beasts of the form
Second, a polynomial of degree 3 or less over any field is irreducible iff it has no roots in the field, so you want that the above quadratic's discriminant has no solution in our field, i.e. : you want that is NOT a quadratic residue .
Well now, just check what elements in are not quadratic residues and count up all the possibilites for ...
For example, as , we get that 5 is not a quadratic residue mod 13 ==> every pair of elements in this field will yield an irreducible quadratic, for example:
** is one irreducible quadratic;
** the quadratic is irreducible...etc.
Of course, there is a formula, but I really don't remember it, though you can look for it in the books...
I understand that the discriminant should not be a quadratic residue but I'm not understanding how you get from to find that 5 is not a quadratic residue.
EDIT: I see. It's the Legendre Symbol. Unfortunately, it hasn't been covered yet in my class and isn't allowed to be used . I'll just find all the quadratic nonresidues by hand by finding all the quadratic residues.