Polynomial Rings

Let be a field. By the ring of polynomials in the indeterminate, written as we mean the set of all symbols



where can be any nonnegative integer and where the coefficients are all in In order to make a ring out of we must be able to recognize when two elements in it are equal, we must be able to add and multiply elements of so that the axioms defining a ring hold true for This will be our initial goal.

We could avoid the phrase ``the set of all symbols'' used above by introducing an appropriate apparatus of sequences but it seems more desirable to follow a path which is somewhat familiar to most readers.

DEFINITION: If and are in then if and only if for every integer

Thus two polynomials are declared to be equal if and only if their corresponding coefficients are equal.

DEFINITION: If and are both in then



where for each

In other words, add two polynomials by adding their coefficients and collecting terms. To add and we consider as and add, according to the recipe given in the definition, to obtain as their sum

The most complicated item, and the only one left for us to define for is the multiplication.

DEFINITION: If and then



where



This definition says nothing more than: multiply the two polynomials by multiplying out the symbols formally, use the relation and collect terms.



DEFINITION: If and then the degree of written as is

That is, the degree of is the largest integer for which the th coefficient of is not 0. We do not define the degree of the zero polynomial. We say a polynomial is a constant if its degree is 0. The degree function defined on the nonzero elements of will provide us with the function needed in order that be a Euclidean ring.

LEMMA 3.9.1: If are two nonzero elements of then




COROLLARY: If are nonzero elements in then


COROLLARY: is an integral domain.

Proof: We leave the proof of this corollary to the reader.

Since is an integral domain, in light of Theorem 3.6.1 we can construct for it its field of quotients. This field merely consists of all quotients of polynomials and is called the field of rational functions in over The function defined for all in satisfies

1. is a nonnegative integer.

2. for all in

In order for to be a Euclidean ring with the degree function acting as the -function of a Euclidean ring we still need that given there exist in such that



where either or This is provided us by

LEMMA 3.9.2 (THE DIVISION ALGORITHM): Given two polynomials and in then there exist two polynomials and in such that



where
or


This last lemma fills the gap needed to exhibit as a Euclidean ring and we now have the right to say

THEOREM 3.9.1: is a Euclidean ring.

All the results of Section 3.7 now carry over and we list these, for our particular case, as the following lemmas. It could be very instructive for the reader to try to prove these directly, adapting the arguments used in Section 3.7 for our particular ring and its Euclidean function, the degree.

LEMMA 3.9.3: is a principal ideal ring.

LEMMA 3.9.4: Given two polynomials in they have a greatest common divisor which can be realized as



What corresponds to a prime element?

DEFINITION: A polynomial in is said to be irreducible over if whenever



with then one of or has degree 0 (i.e., is a constant).

Irreducibility depends on the field; for instance the polynomial is irreducible over the real field but not over the complex field, for there



where

LEMMA 3.9.5: Any polynomial in can be written in a unique manner as a product of irreducible polynomials in

LEMMA 3.9.6: The ideal in is a maximal ideal if and only is irreducible over

In Chapter 5 we shall return to take a much closer look at this field but for now we should like to compute an example.