Counting Number of Irreducibles

[e(ho0n3]

Homework Statement
Let p be a prime.
(a) Determine the number of irreducible polynomials over Z_p of the form x² + ax + b.
(b) Determine the number of irreducible quadratic polynomials over Z_p.

The attempt at a solution
A nonzero, nonunit polynomial f(x) in Z_p[x] is irreducible if it equals the product of two polynomials in Z_p[x], one of them being a unit of Z_p[x]. But does Z_p[x] have any units? I find it hard to imagine that there are polynomials g(x) and h(x) in Z_p[x] with g(x)h(x) = 1, unless of course g(x) = h(x) = ±1.

 

[matt grime]

I find it hard to imagine that there are polynomials g(x) and h(x) in Z_p[x] with g(x)h(x) = 1.

just considering the leading terms of g and h will show you this is a trivial statement.

You could also try counting the reducible quadratics, since that is also easy.

 

[e(ho0n3]

I think I understand: Let ax^m and bxⁿ be the leading terms of g(x) and h(x). If g(x)h(x) = 1, then abx^m+n must equal 0 so ab must equal p but this is impossible. It follows that Z_p[x] has no units so the answer both (a) and (b) is 0?

 

[e(ho0n3]

Oh, and how does counting the reducible quadratics help in counting the irreducible ones?

 

[matt grime]

How long did you think about that (the counting irreducibles)?

 

[e(ho0n3]

Not long. However, something just occurred to me: There are p³ polynomials in Z_p[x] so the number of irreducibles is just p³ minus the number of reducibles. Is that right?

 

[matt grime]

There are *not* p^3 polys in Z_p[x] - there may be p^3 in some subset of that. But it is certainly true that a poly is either reducible or irreducible, and not both, so if you know how many in total there are, and how many of one type, then you know how many of the other there are.

And there are not p^3 quadratics, if that was what you meant, since the leading coefficient has to be non-zero.

 

[e(ho0n3]

Yes, I did mean p³ quadratics. Sorry about that. So would (p - 1)p² be a better estimate?

I'm confused though: If Z_p[x] doesn't have any units (cf. my second post), how can there be irreducible polynomials?

 

[matt grime]

In what way does x not satisfy the definition of irreducible? How are you going to write that as a product of two other polynomials of strictly smaller degree. And why are there no units? You even wrote out 2 units in your first post.

 

[e(ho0n3]

x is irreducible since the only way to write it as a product g(x)h(x) is by setting g(x) = 1 and h(x) = x or vice-versa and 1 is a unit. I understand that.

When I wrote "Z_p[x] doesn't have any units", I meant nonconstant units. I can't think of an example unit in Z_p[x] that is nonconstant. This is probably why I wrote that there aren't any.