a2 b2 = a2 b2 . Then we have

(a1 b2 + a2 b1 )b1 b2 = a1 b2 b1 b2 + a2 b1 b1 b2 = a1 b2 b1 b2 + a2 b1 b1 b2

= (a1 b2 + a2 b1 )b1 b2

and

a1 a2 b1 b2 = a1 a2 b1 b2 = a1 a2 b1 b2 . 2

In light of this lemma, we may unambiguously de¬ne addition and multi-

plication on K as follows: for [a1 , b1 ], [a2 , b2 ] ∈ K, we de¬ne

[a1 , b1 ] + [a2 , b2 ] := [a1 b2 + a2 b1 , b1 b2 ]

and

[a1 , b1 ] · [a2 , b2 ] := [a1 a2 , b1 b2 ].

The next task is to show that K is a ring ” we leave the details of this

(which are quite straightforward) to the reader.

Lemma 17.4. With addition and multiplication as de¬ned above, K is a

ring, with additive identity [0D , 1D ] and multiplicative identity [1D , 1D ].

17.2 The ¬eld of fractions of an integral domain 365

Proof. Exercise. 2

Finally, we observe that K is in fact a ¬eld: it is clear that [a, b] is a non-

zero element of K if and only if a = 0D , and hence any non-zero element

[a, b] of K has a multiplicative inverse, namely, [b, a].

The ¬eld K is called the ¬eld of fractions of D. Consider the map

„ : D ’ K that sends a ∈ D to [a, 1D ] ∈ K. It is easy to see that this map

is a ring homomorphism, and one can also easily verify that it is injective.

So, starting from D, we can synthesize “out of thin air” its ¬eld of fractions

K, which essentially contains D as a subring, via the canonical embedding

„ : D ’ K.

Now suppose that we are given a ¬eld L that contains D as a subring.

Consider the set K consisting of all elements in L of the form ab’1 , where

a, b ∈ D and b = 0”note that here, the arithmetic operations are performed

using the rules for arithmetic in L. One may easily verify that K is a

sub¬eld of L that contains D, and it is easy to see that this is the smallest

sub¬eld of L that contains D. The sub¬eld K of L may be referred to as

the ¬eld of fractions of D within L. One may easily verify that the map

ρ : K ’ L that sends [a, b] ∈ K to ab’1 ∈ L is an unambiguously de¬ned

ring homomorphism that maps K injectively onto K ; in particular, K is

isomorphic as a ring to K . It is in this sense that the ¬eld of fractions K is

the smallest ¬eld containing D as a subring.

Somewhat more generally, suppose that L is a ¬eld, and that „ : D ’ L

is an embedding. One may easily verify that the map ρ : K ’ L that sends

[a, b] ∈ K to „ (a)„ (b)’1 ∈ L is an unambiguously de¬ned, injective ring

homomorphism. Moreover, we may view K and L as D-algebras, via the

embeddings „ : D ’ K and „ : D ’ L, and the map ρ is seen to be a

D-algebra homomorphism.

From now on, we shall simply write an element [a, b] of K as a fraction,

a/b. In this notation, the above rules for addition, multiplication, and testing

equality in K now look quite familiar:

a1 a2 a1 b2 + a2 b1 a1 a2 a1 a2 a1 a2

·

+ = , = , and = i¬ a1 b2 = a2 b1 .

b1 b2 b1 b2 b 1 b2 b1 b2 b1 b2

Observe that for a, b ∈ D, with b ∈ 0D and b | a, so that a = bc for

some c ∈ D, then the fraction a/b ∈ K is equal to the fraction c/1D ∈ K,

and identifying the element c ∈ D with its canonical image c/1D ∈ K, we

may simply write c = a/b. Note that this notation is consistent with that

introduced in part (iii) of Theorem 9.4. A special case of this arises when

b ∈ D— , in which case c = ab’1 .

366 More rings

Function ¬elds

An important special case of the above construction for the ¬eld of fractions

of D is when D = F [X], where F is a ¬eld. In this case, the ¬eld of fractions is

denoted F (X), and is called the ¬eld of rational functions (over F ). This

terminology is a bit unfortunate, since just as with polynomials, although

the elements of F (X) de¬ne functions, they are not (in general) in one-to-one

correspondence with these functions.

Since F [X] is a subring of F (X), and since F is a subring of F [X], we see

that F is a sub¬eld of F (X).

More generally, we may apply the above construction to the ring D =

F [X1 , . . . , Xn ] of multi-variate polynomials over a ¬eld F , in which case the

¬eld of fractions is denoted F (X1 , . . . , Xn ), and is also called the ¬eld of

rational functions (over F , in the variables X1 , . . . , Xn ).

Exercise 17.5. Let F be a ¬eld of characteristic zero. Show that F contains

an isomorphic copy of Q.

Exercise 17.6. Show that the ¬eld of fractions of Z[i] within C is Q[i]. (See

Example 9.22 and Exercise 9.8.)

17.3 Unique factorization of polynomials

Throughout this section, F denotes a ¬eld.

Like the ring Z, the ring F [X] of polynomials is an integral domain, and

because of the division with remainder property for polynomials, F [X] has

many other properties in common with Z. Indeed, essentially all the ideas

and results from Chapter 1 can be carried over almost verbatim from Z to

F [X], and in this section, we shall do just that.

Recall that for a, b ∈ F [X], we write b | a if a = bc for some c ∈ F [X], and

in this case, note that deg(a) = deg(b) + deg(c).

The units of F [X] are precisely the units F — of F , that is, the non-zero

constants. We call two polynomials a, b ∈ F [X] associate if a = ub for

u ∈ F — . It is easy to see that a and b are associate if and only if a | b

and b | a”indeed, this follows as a special case of part (ii) of Theorem 9.4.

Clearly, any non-zero polynomial a is associate to a unique monic polynomial

(i.e., with leading coe¬cient 1), called the monic associate of a; indeed,

the monic associate of a is lc(a)’1 · a.

We call a polynomial p irreducible if it is non-constant and all divisors

of p are associate to 1 or p. Conversely, we call a polynomial n reducible

if it is non-constant and is not irreducible. Equivalently, non-constant n is

17.3 Unique factorization of polynomials 367

reducible if and only if there exist polynomials a, b ∈ F [X] of degree strictly

less that n such that n = ab.

Clearly, if a and b are associate polynomials, then a is irreducible if and

only if b is irreducible.

The irreducible polynomials play a role similar to that of the prime num-

bers. Just as it is convenient to work with only positive prime numbers, it

is also convenient to restrict attention to monic irreducible polynomials.

Corresponding to Theorem 1.3, every non-zero polynomial can be ex-

pressed as a unit times a product of monic irreducibles in an essentially

unique way:

Theorem 17.5. Every non-zero polynomial n ∈ F [X] can be expressed as

n = u · pe1 · · · per ,

r

1

where u ∈ F — , the pi are distinct monic irreducible polynomials, and the ei

are positive integers. Moreover, this expression is unique, up to a reordering

of the pi .

To prove this theorem, we may assume that n is monic, since the non-

monic case trivially reduces to the monic case.

The proof of the existence part of Theorem 17.5 is just as for Theorem 1.3.

If n is 1 or a monic irreducible, we are done. Otherwise, there exist a, b ∈

F [X] of degree strictly less than n such that n = ab, and again, we may

assume that a and b are monic. By induction on degree, both a and b can

be expressed as a product of monic irreducible polynomials, and hence, so

can n.

The proof of the uniqueness part of Theorem 17.5 is almost identical

to that of Theorem 1.3. As a special case of Theorem 9.12, we have the

following division with remainder property, analogous to Theorem 1.4:

Theorem 17.6. For a, b ∈ F [X] with b = 0, there exist unique q, r ∈ F [X]

such that a = bq + r and deg(r) < deg(b).

Analogous to Theorem 1.5, we have:

Theorem 17.7. For any ideal I ⊆ F [X], there exists a unique polynomial d

such that I = dF [X], where d is either zero or monic.

Proof. We ¬rst prove the existence part of the theorem. If I = {0}, then

d = 0 does the job, so let us assume that I = {0}. Let d be a monic

polynomial of minimal degree in I. We want to show that I = dF [X].

We ¬rst show that I ⊆ dF [X]. To this end, let c be any element in I. It

368 More rings

su¬ces to show that d | c. Using Theorem 17.6, we may write c = qd + r,