All the results of Section 1.2 on divisibility and greatest common divisors in \(\mathbb{Z}\) now carry over, with only minor modifications, to the ring of polynomials over a field. Throughout this section, \(F\) always denotes a field.

Definition

Let \( F \) be a field and \( a(x), b(x) \in F[x] \) with \( b(x) \) nonzero. We say that \( b(x) \) divides \( a(x) \) [or that \( b(x) \) is a factor of \( a(x) \)], and write \( b(x) | a(x) \) if \( a(x) = b(x)h(x) \) for some \( h(x) \in F[x] \).

Example 1

\((2x + 1)(6x^2 - x - 2) \in \mathbb{Q}[x]\) because \(6x^2 - x - 2 = (2x + 1)(3x - 2)\).

Furthermore, every constant multiple of \(2x + 1\) also divides \(6x^2 - x - 2\). For instance, \[ 5(2x + 1) = 10x + 5 \text{ divides } 6x^2 - x - 2 \text{ because } 6x^2 - x - 2 = 5(2x + 1)\left(\frac{3x - 2}{5}\right). \]

Example 1 illustrates the first part of the following result.

Theorem 4.7

Let \( F \) be a field and \( a(x), b(x) \in F[x] \) with \( b(x) \) nonzero.

1. If \( b(x) \) divides \( a(x) \), then \( c b(x) \) divides \( a(x) \) for each nonzero \( c \in F \).
2. Every divisor of \( a(x) \) has degree less than or equal to \( \text{deg}(a(x)) \).

Proof

1. If \( b(x) \) divides \( a(x) \), then \( a(x) = b(x)h(x) \) for some \( h(x) \in F[x] \). Hence, \( a(x) = 1_F \cdot b(x)h(x) = c \cdot c^{-1}b(x)h(x) = (cb(x))(c^{-1}h(x)) \). Therefore, \( cb(x) \) divides \( a(x) \).

2. Suppose \( b(x) | a(x) \), say \( a(x) = b(x)h(x) \). By Theorem 4.2, \[ \text{deg}(a(x)) = \text{deg}(b(x)) + \text{deg}(h(x)). \]

Since degrees are nonnegative, we must have \( 0 \leq \text{deg}(b(x)) \leq \text{deg}(a(x)) \). \(\blacksquare\)

As we learned earlier, the greatest common divisor of two integers is the largest integer that divides both of them. By analogy, the greatest common divisor of two polynomials \( a(x), b(x) \in F[x] \) ought to be the polynomial of highest degree that divides both of them. But such a greatest common divisor would not be unique because each constant multiple of it would have the same degree and would also divide both \( a(x) \) and \( b(x) \). In order to guarantee a unique gcd, we modify this definition slightly by introducing a new concept. A polynomial in \( F[x] \) is said to be monic if its leading coefficient is 1. For instance, \( x^3 + x + 2 \) is monic in \( \mathbb{Q}[x] \), but \( 2x + 1 \) is not.

Definition

Let \( F \) be a field and \( a(x), b(x) \in F[x] \), not both zero. The greatest common divisor (gcd) of \( a(x) \) and \( b(x) \) is the monic polynomial of highest degree that divides both \( a(x) \) and \( b(x) \).

1. \( d(x) | a(x) \) and \( d(x) | b(x) \) is the gcd of \( a(x) \) and \( b(x) \) provided that \( d(x) \) is monic and of highest degree.
2. If \( a(x) | c(x) \) and \( b(x) | c(x) \), then \( \text{deg}(c(x)) \geq \text{deg}(d(x)) \).

Polynomials \( a(x) \) and \( b(x) \) have at least one monic common divisor (namely \( 1_F \)). Since the degree of a common divisor of \( a(x) \) and \( b(x) \) cannot exceed either \( \text{deg}(a(x)) \) or \( \text{deg}(b(x)) \) by Theorem 4.7, there must be at least one monic common divisor of highest degree. In Theorem 4.8 below, we shall show that there is only one monic common divisor of highest degree, thus justifying the definition’s reference to the greatest common divisor.

Example 2

To find the gcd of \( 3x^2 + x + 6 \) and \( 0 \) in \( \mathbb{Q}[x] \), we note that the common divisors of highest degree are just the divisors of \( 3x^2 + x + 6 \) of degree 2. These include \( 3x^2 + x + 6 \) itself and all nonzero constant multiples of this polynomial—in particular, the monic polynomial \[ \frac{1}{3}(3x^2 + x + 6) = x^2 + \frac{1}{3}x + 2. \]

Hence, \( x^2 + \frac{1}{3}x + 2 \) is a gcd of \( 3x^2 + x + 6 \) and \( 0 \).

Example 3

You can easily verify these factorizations in \( \mathbb{Q}[x] \):

\[ a(x) = 2x^4 + 5x^3 - 5x - 2 = (2x + 1)(x^3 + 2x - 1)(x - 1), \] \[ b(x) = 2x^3 - 3x^2 - 2x = (2x + 1)(x^2 - 2x - 1). \]

It appears that \( 2x + 1 \) is a common divisor of highest degree of \( a(x) \) and \( b(x) \). The constant multiple \( \frac{1}{2}(2x + 1) \) is a monic common divisor of highest degree. For a proof that \( x + \frac{1}{2} \) actually is the greatest common divisor, see Exercise 5(g).

The remainder of this section, which is referred to only a few times in the rest of the book, may be skimmed if time is short—read the theorems and corollaries, but skip the proofs.

Theorem 4.8

Let \( F \) be a field and \( a(x), b(x) \in F[x] \), not both zero. Then there is a unique greatest common divisor \( d(x) \) of \( a(x) \) and \( b(x) \). Furthermore, there are (not necessarily unique) polynomials \( u(x) \) and \( v(x) \) such that \( d(x) = a(x)u(x) + b(x)v(x) \).

Steps 1 and 2 of the proof are patterned after the proof of Theorem 1.2.

Proof of Theorem 4.8

Let \( S \) be the set of all linear combinations of \( a(x) \) and \( b(x) \), that is, \[ S = \{a(x)m(x) + b(x)n(x) | m(x), n(x) \in F[x]\}. \]

Step 1: Find a monic polynomial of smallest degree in \( S \).

Proof of Step 1: \( S \) contains nonzero polynomials (for instance, at least one of \( a(x) \cdot 1_F + b(x) \cdot 0_F \) or \( a(x) \cdot 0_F + b(x) \cdot 1_F \)). So the set of all the degrees of polynomials in \( S \) is a nonempty set of nonnegative integers, which has a smallest element by the Well-Ordering Axiom. Hence, there is a polynomial \( u(x) \) of smallest degree in \( S \). If \( d \) is the leading coefficient of \( u(x) \), then \( d^{-1}u(x) \) is a monic polynomial of smallest degree in \( S \). By the definition of \( S \), \[ t(x) = a(x)u(x) + b(x)v(x) \text{ for some } u(x), v(x) \in F[x]. \]

Step 2: Prove that \( t(x) \) is a gcd of \( a(x) \) and \( b(x) \).

Proof of Step 2: We must prove that \( t \) satisfies the two conditions in the definition of the gcd:

1. \( t(x) | a(x) \) and \( t(x) | b(x) \),
2. If \( c(x) | a(x) \) and \( c(x) | b(x) \), then \( \text{deg}(c(x)) \leq \text{deg}(t(x)) \).

Proof of (1): In the proof of Step 2 of Theorem 1.2, replace \( a, b, c, t, q, r, u, v, k, \) and \( s \) with \( a(x), b(x), c(x), t(x), u(x), v(x), k(x), \) and \( s(x) \), respectively, to show that \( t(x) \) is a common divisor of \( a(x) \) and \( b(x) \).

Proof of (2): With the same replacements as in the proof of (1), repeat the proof of Step 2 of Theorem 1.2, until you reach this statement: \[ t(x) = a(x)u(x) + b(x)v(x) = [c(x)k(x)]u(x) + [c(x)s(x)]v(x). \]

The first and last terms of this equation show that \( c(x) | t(x) \). By Theorem 4.7, \( \text{deg}(c(x)) \leq \text{deg}(t(x)) \).

This shows that \( t(x) \) is a greatest common divisor of \( a(x) \) and \( b(x) \).

Step 3: Prove that \( t(x) \) is the unique gcd of \( a(x) \) and \( b(x) \).

Proof of Step 3: Suppose that \( d(x) \) is any gcd of \( a(x) \) and \( b(x) \). To prove uniqueness, we must show that \( d(x) = t(x) \). Since \( d(x) \) is a common divisor, we have \[ a(x) = d(x)f(x) \text{ and } b(x) = d(x)g(x) \text{ for some } f(x), g(x) \in F[x]. \]

Therefore, \[ t(x) = a(x)u(x) + b(x)v(x) = [d(x)f(x)]u(x) + [d(x)g(x)]v(x) = d(x)[f(x)u(x) + g(x)v(x)]. \]

By Theorem 4.2, \[ \text{deg}(t(x)) = \text{deg}(d(x)) + \text{deg}[f(x)u(x) + g(x)v(x)]. \]

Since they are gcd's, \( t(x) \) and \( d(x) \) have the same degree. Hence, \[ \text{deg}[f(x)u(x) + g(x)v(x)] = 0, \]

so that \( f(x)u(x) + g(x)v(x) = c \) for some constant \( c \in F \). Therefore, \( t(x) = c d(x) \). Since both \( t(x) \) and \( d(x) \) are monic, the leading coefficient on the left side is \( 1_F \), and the leading coefficient on the right side is \( c \). So we must have \( c = 1_F \). Therefore, \( t(x) = d(x) \), and \( t(x) \) is the unique gcd of \( a(x) \) and \( b(x) \). \(\blacksquare\)

Corollary 4.9

Let \( F \) be a field and \( a(x), b(x) \in F[x] \), not both zero. A monic polynomial \( d(x) \in F[x] \) is the greatest common divisor of \( a(x) \) and \( b(x) \) if and only if \( d(x) \) satisfies these conditions.

1. \( d(x) | a(x) \) and \( d(x) | b(x) \).
2. If \( c(x) | a(x) \) and \( c(x) | b(x) \), then \( c(x) | d(x) \).

Proof: Adapt the proof of Corollary 1.3 to \( F[x] \).

Polynomials \( f(x) \) and \( g(x) \) are said to be relatively prime if their greatest common divisor is \( 1_F \).

Theorem 4.10

Let \( F \) be a field and \( a(x), b(x), c(x) \in F[x] \). If \( a(x) | b(x)c(x) \) and \( a(x) \) and \( b(x) \) are relatively prime, then \( a(x) | c(x) \).

Proof: Adapt the proof of Theorem 1.4 to \( F[x] \). \(\blacksquare\)