A polynomials with coefficients in a field

[R.P.F.]

Homework Statement

Prove that a polynomial f of degree n with coefficients in a field F has at most n roots in F.

Homework Equations

The Attempt at a Solution

So we could prove this by induction by using a is a root of f if and only if x-a divides f. My question is: why do the coefficients have to be in a field? Shouldn't an integral domain work? Thanks!

 

[Hurkyl]

Once you've proven it for a field, it's easy to prove it for an integral domain.

If you tried to prove it directly for integral domains, it would be trickier -- the field case has fewer details to worry about.

 

[R.P.F.]

Once you've proven it for a field, it's easy to prove it for an integral domain.

If you tried to prove it directly for integral domains, it would be trickier -- the field case has fewer details to worry about.

Hmmm...Doesn't the same proof work for an integral domain? I mean the claim that
a is a root of f(x) in R[x] if and only if x-a divides f(x) is true as long as R is an integral domain(or maybe as long as R is a ring?). Then we can just use induction.

 

[Hurkyl]

It's not too hard to find a counterexample to the case of general rings.

For example, consider the polynomial x² - 1 with coefficients in the ring of integers modulo 8. It factors in two ways:

x²-1 = (x-1)(x-7) = (x-3)(x-5)​