Polynomial Rings

• If R is a ring, then , the ring of polynomials in x with coefficients in R, consists of all formal sums , where for all but finitely many values of i.
• If is a nonzero polynomial, the degree is the largest such that . The zero polynomial has degree .
• becomes a ring with the usual operations of polynomial addition and multiplication.
• If F is a field, the units in are exactly the nonzero elements of F.
• If is a field, , and , there are unique polynomials such that

• Let R be an integral domain. An element is irreducible if , x is not a unit, and if implies either y is a unit or z is a unit.
• Let R be an integral domain. An element is prime if , x is not a unit, and implies or .
• In an integral domain, primes are irreducible.
• Let F be a field. If are not both zero, then and have a greatest common divisor which is unique up to multiplication by units (elements of F).
• Let F be a field, let , and let be a greatest common divisor of and . There are polynomials such that

If R is a ring, the ring of polynomials in x with coefficients in R is denoted . It consists of all formal sums

Here for all but finitely many values of i.

If the idea of "formal sums" worries you, replace a formal sum with the infinite vector whose components are the coefficients of the sum:

All of the operations which I'll define using formal sums can be defined using vectors. But it's traditional to represent polynomials as formal sums, so this is what I'll do.

A nonzero polynomial has degree n if and , and n is the largest integer with this property. The zero polynomial is defined by convention to have degree . (This is necessary in order to make the dgree formulas work out.) Alternatively, you can say that the degree of the zero polynomial is undefined; in that case, you will need to make minor changes to some of the results below.

Polynomials are added componentwise, and multiplied using the "convolution" formula:

These formulas say that you compute sums and products as usual.

Example. ( Polynomial arithmetic) In ,

Let R be an integral domain. Then If , write to denote the degree of f. It's easy to show that the degree function satisfies the following properties:

The verifications amount to writing out the formal sums, with a little attention paid to the case of the zero polynomial. These formulas {\it do} work if either f or g is equal to the zero polynomial, provided that is understood to behave in the obvious ways (e.g. for any ).

Example. ( Degrees of polynomials) Note that in ,

This shows that equality might not hold in .

The equality might not hold if R is not an integral domain. For example, take . Then (since ),

Lemma. Let F be a field, and let be the polynomial ring in one variable over F. The units in are exactly the nonzero elements of F.

Proof. It's clear that the nonzero elements of F are invertible in , since they're already invertible in F. Conversely, suppose that is invertible, so for some . Then , which is impossible unless f and g both have degree 0. In particular, f is a nonzero constant, i.e. an element of F.

Theorem. ( Division Algorithm) Let F be a field, and let . Suppose that . There exist such that

Proof. The idea is to imitate the proof of the Division Algorithm for .

Let

The set is a subset of the nonnegative integers, and therefore must contain a smallest element by well-ordering. Let be an element in S of smallest degree, and write

I need to show that .

If , then since , .

Suppose then that . Assume toward a contradiction that . Write

Assume , and .

Consider the polynomial

Its degree is less than n, since the n-th degree terms cancel out.

However,

The latter is an element of S.

I've found an element of S of smaller degree than , which is a contradiction. It follows that , and this completes the proof.

Example. ( Polynomial division) Division of polynomials should be familiar to you --- at least over and .

In this example, I'll divide by in . Remember as you follow the division that , , and --- I'm doing arithmetic mod 5.

If you prefer, you can do long division without writing the powers of x --- i.e. just writing down the coefficients. Here's how it looks:

Either way, the quotient is and the remainder is :

Definition. Let R be a commutative ring and let . An element is a root of if .

Note that polynomials are actually formal sums, not functions. However, it is obvious how to plug a number into a polynomial. Specifically, let

For , define

Observe that a polynomial can be nonzero as a polynomial even if it equals 0 for every input! For example, take is a nonzero polynomial. However, plugging in the two elements of the coefficient ring gives

Corollary. Let F be a field, and let , where .

(a) ( The Root Theorem) c is a root of in F if and only if .

(b) has at most n roots in F.

Proof. (a) Suppose . Write

Then or .

In the first case, r is a nonzero constant. However, this implies that

This contradiction shows that , and .

Conversely, if is a factor of , then for some . Hence,

and c is a root of f.

(b) If are the distinct roots of f in F, then

Taking degrees on both sides gives .

Example. ( Applying the Root Theorem) Consider polynomials in .

If , it's obvious that is a root. Therefore, is a factor of .

Likewise, must be a factor of for any , since is a root of .

Example. ( Applying the Root Theorem) Prove that is divisible by in .

Plugging in gives

Since is a root, is a factor, by the Root Theorem.

Example. ( A polynomial with more roots than its degree) The quadratic polynomial has roots , , , . The previous result does not apply, because is not a field.

On the other hand, has at most 3 roots over , since is a field. (In fact, this polynomial has no roots in , as you can verify by plugging in 0, 1, 2, 3, and 4.

Corollary. ( The Remainder Theorem) Let F be a field, , and let . When is divided by , the remainder is .

Proof. Divide by :

Since , it follows that is a constant. But

Therefore, the constant value of is .

Example. ( Applying the Remainder Theorem) Suppose leaves a remainder of 5 when divided by and a remainder of -1 when divided by . What is the remainder when is divided by ?

By the second Corollary above,

Now divide by . The remainder has degree less than , so for some :

Then

Solving the two equations for a and b, I get and . Thus, the remainder is .

Definition. Let R be an integral domain.

(a) If , then x divides y if for some . means x divides y.

(b) x and y are associates if , where u is a unit. (Recall that a unit in a ring is an element with a multiplicative inverse.)

(c) An element is irreducible if , x is not a unit, and if implies either y is a unit or z is a unit.

(d) An element is prime if , x is not a unit, and implies or .

Example. ( Irreducible polynomials) A nonzero polynomial is irreducible if and only if implies that either g or h is a constant.

For example, is irreducible in ; it is not irreducible in , because in .

Corollary. Let F be a field. A polynomial of degree 2 or 3 in is irreducible if and only if it has no roots in F.

Proof. Suppose has degree 2 or 3.

If f is not irreducible, then , where neither g nor h is constant. Now and ; since

this is only possible if at least one of g or h has degree 1. This means that at least one of g or h is a linear factor , and must therefore have a root in F. Since , it follows that f has a root in F as well.

Conversely, if f has a root c in F, then is a factor of f by the Root Theorem. Since f has degree 2 or 3, is a proper factor, and f is not irreducible.

Example. ( Checking for irreducibility of a quadratic or cubic) Consider . Since has no roots in , is irreducible.

On the other hand, consider .

Since has no roots in , it's irreducible.

Example. ( Roots and irreducibility for polynomials of degree bigger than 3) If is a polynomial of degree greater than 3, the fact that f has no roots does not mean that it's irreducible.

For example, has no roots in , but it is surely not irreducible over .

What about ? Does it factor over ?

Lemma. In an integral domain, primes are irreducible.

Proof. Let x be prime. I must show x is irreducible. Suppose . I must show either y or z is a unit.

, so obviously . Thus, or . Without loss of generality, suppose .

Write . Then , and since (primes are nonzero) and we're in a domain, . Therefore, z is a unit, and x is irreducible.

Definition. Let R be an integral domain, and let . is a greatest common divisor of x and y if:

(a) and .

(b) If and , then .

The definition says "a" greatest common divisor, rather than "the" greatest common divisor, because greatest common divisors are only unique up to multiplication by units.

The definition above is the right one if you're dealing with an arbitrary integral domain. However, if your ring is a polynomial ring, it's nice to single out a "special" greatest common divisor and call it the greatest common divisor.

Definition. A monic polynomial is a polynomial whose leading coefficient is 1.

For example, here are some monic polynomials over :

Definition. Let F be a field, let be the ring of polynomials with coefficients in F, and let , where f and g are not both zero. The greatest common divisor of f and g is the monic polynomial which is a greatest common divisor of f and g (in the integral domain sense).

Example. ( Polynomial greatest common divisors) Consider the polynomial ring . is a greatest common divisor of and :

Notice that any nonzero constant multiple of is also a greatest common divisor of and (in the integral domain sense): For example, works. This makes sense, because the units in are the nonzero elements of . But by convention, I'll refer to --- the monic greatest common divisor --- as the greatest common divisor of and .

The preceding definition assumes there is a greatest common divisor for two polynomials in . In fact, the greatest common divisor of two polynomials exists --- provided that both polynomials aren't 0 --- and the proof is essentially the same as the proof for greatest common divisors of integers.

In both cases, the idea is to use the Division Algorithm repeatedly until you obtain a remainder of 0. This must happen in the polynomial case, because the Division Algorithm for polynomials specifies that the remainder has strictly smaller degree than the divisor.

Just as in the case of the integers, each use of the Division Algorithm does not change the greatest common divisor. So the last pair has the same greatest common divisor as the first pair --- but the last pair consists of 0 and the last nonzero remainder, so the last nonzero remainder is the greatest common divisor.

This process is called the Euclidean algorithm, just as in the case of the integers.

Let h and be two greatest common divisors of f and g. By definition, and . From this, it follows that h and have the same degree, and are constant multiples of one another. If h and are both monic --- i.e. both have leading coefficient 1 --- this is only possible if they're equal. So there is a unique monic greatest common divisor for any two polynomials.

Finally, the same proofs that I gave for the integers show that you can write the greatest common divisor of two polynomials as a linear combination of the two polynomials. You can use the Extended Euclidean Algorithm that you learned for integers to find a linear combination. To summarize:

Theorem. Let F be a field, , f and g not both 0.

(a) f and g have a unique (monic) greatest common divisor.

(b) There exist polynomials such that

Example. ( Applying the Extended Euclidean Algorithm to polynomials) You can use the tabular method of the Extended Euclidean Algorithm to express the greatest common divisor of two polynomials as a linear combination of the two polynomials. For example, consider and in .

The greatest common divisor is . The greatest common divisor is only determined up to multiplying by a unit, so multiplying by gives the monic greatest common divisor .

You can check that

Example. ( Applying the Extended Euclidean Algorithm to polynomials over ) Find the greatest common divisor of and in and express the greatest common divisor as a linear combination of and with coefficients in .

The greatest common divisor is , and

The greatest common divisor is only determined up to multiplying by a unit. So, for example, I can multiply the last equation by 2 to get

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