#polynomial ring

The truth is, while we may build factor rings, what people really do a lot more is build up rings of polynomials. Let's start with a normal ring, R. We can create a new ring made up of polynomials whose coefficients come from R, usually using the placeholder indeterminant x. We call this ring R[x], or R adjoint x. (We can just as well build rings of polynomials with more than one indeterminant, R[x, y, z, ...], but that's just a natural extension of the single-indeterminant polynomial ring stuff.)

So in this new ring, our elements are polynomials with coefficients from R. (I know I talked a little bit about solving polynomials, but right now the coefficients are more in the spotlight than the indeterminant.) Addition of elements in R[x] is just done by adding coefficients of the same order of indeterminant, using addition from R. Multiplication is done using the distribution (distributive property) and multiplication from R. It's kind of a simple thing to notice, but I'm going to say it anyway: R is embedded in R[x] in the form of constant polynomials.

Like with factor rings, the original additional structure of a ring may not carry over after this kind of fiddling. For example, we can start with a field, say Q. Then we may guess Q[x] is a field as well. In fact, it is definitely not a field, since the polynomial p(x) = x does not have an inverse (notice this remark is true for any ring of polynomials). However, the integral domain structure of a ring will carry over: if D is an integral domain, then D[x] will still be an integral domain.

Presumably, we've seen polynomials for a pretty long time by now...probably starting since middle school. But now we're thinking about them in a slightly different way--they're mathematical objects with coefficients from algebraic ring structures. They aren't just equations with numbers. That said, we need to be a little more careful with our language. In particular, we can talk about factoring polynomials, but we have to specify over what ring we are factoring the polynomial (ie, where we are drawing the coefficients from). For example, we cannot factor x2 - 2 over Q, but we can factor it over R.

We like to think about factoring because it helps us solve polynomials for zeros. While that trick was pretty useful in the past, it grinds to a stop when we're in a ring with zero divisors. Despite completely factoring a polynomial in such a ring, it's no guarantee that we've found all the zeros. Even integral domains make things tricky because factoring is hard without having all possible inverses. So in the interest of making things easier for ourselves, we're going to think about polynomials over fields from now on.

There follows a whole bunch of uninteresting stuff about factoring polynomials after this, but the main gist is that, given a polynomial p(x) over a field with degree higher than q(x), there is a unique factoring p(x) = q(x)s(x) + r(x) where the degree of r(x) is lower than the degree of q(x). If s(x) = 0, then the polynomial is irreducible with respect to q(x).

A polynomial is irreducible over R if it cannot be expressed as the product of two lower-degree polynomials with coefficients from R. Again, we need to be careful with language here. When talking about irreducibility, we unfortunately need to be pedantic and specify exactly which ring (or field) we are trying to factor the polynomial over.

Why all the hooplah about irreducible polynomials and factoring polynomials in general? A few reasons. We've already said that factoring helps find zeros. In fact, over fields, (x - a) is a factor of a polynomial iff it is a zero of the polynomial. As far as irreducible polynomials go, they are kind of like the prime numbers of the polynomial world. They also building up the maximal ideals in F[x].

The first observation is that all ideals in F[x] are principal. The second observation is that ideals in F[x] are maximal iff the generating polynomial is irreducible over F. We care about maximal ideals in F[x] because, as stated above, polynomial rings are never fields. In order to build fields from them, we'd like to be able to mod out by a maximal ideal.