Divisibility in F[x]

If we review the first chapter, after proving the division algorithm for integers, we used it to establish many properties regrading divisibility of integers. In this section, we will do the same for polynomials.

Almost all of the statements carries over to polynomials with the similar proofs. We will not repeat the proofs here, but we will state the results and give examples.

We will fix a field $F$ and all polynomial will be taken from $F[x]$.

Units in $F[x]$

Before we state results for divisibility, it is important to understand what are the units in $F[x]$. The units in $\mathbb{Z}$ are $\pm 1$, we will have more units in $F[x]$.

Proposition

A polynomial $f(x)$ is a unit in $F[x]$ if and only if $f(x)=a$ for some $a\in F\setminus \{0_F\}$.

proof of the proposition

If $f(x)$ is a unit, then there exists a polynomial $g(x)$ such that $f(x)g(x)=1$. Since the degree of the product of two polynomials is the sum of the degrees of the polynomials, we must have that $f(x)$ and $g(x)$ are both constant polynomials. Therefore, $f(x)=a$ for some $a\in F$, and $g(x)=b$ for some $b\in F$. Since $f(x)g(x)=1$, we must have that $a\cdot b=1$. Therefore, $a$ is a unit in $F$, i.e. $a\in F\setminus \{0_F\}$.

The converse is trivial, since $a$ is a unit in $F$, we can take $g(x)=a^{-1}$.

Divisors and Greatest Common Divisors

We start with the definition of divisibility for polynomials.

Definition

A polynomial $f(x)$ is said to divide a polynomial $g(x)$ if there exists a polynomial $h(x)$ such that $g(x)=f(x)h(x)$. We will denote this by $f(x)\mid g(x)$, $f(x)$ is called a divisor of $g(x)$.

Divisibility is not affected by multiplication by a unit.

Proposition

If $f(x)\mid g(x)$, then $cf(x)\mid g(x)$ for any $c\in F\setminus \{0_F\}$.

Since we can always rescale a polynomial by a unit, we will often assume a divisor will have the leading coefficient of the polynomial is $1_F$. We give a special name to such polynomials.

Definition

A polynomial $f(x)$ is called a monic polynomial if the leading coefficient of $f(x)$ is $1_F$.

Again, any polynomial divides $0_F$.

Proposition

For any polynomial $f(x)$, $f(x)\mid 0_F$.

For non-zero $g(x)$, a divisor $f(x)$ will have degree less than or equal to the degree of $g(x)$.

Proposition

If $f(x)\mid g(x)$, and $g(x)\ne 0_F$, then $\mathrm{deg}(f(x))\le \mathrm{deg}(g(x))$.

We can define common divisors and greatest common divisors in the same way as we did for integers.

Definition

A polynomial $d(x)$ is called a common divisor of $f(x)$ and $g(x)$ if $d(x)\mid f(x)$ and $d(x)\mid g(x)$.

To define greatest common divisors, we need a metric to compare polynomials, obviously the degree of the polynomial is a good metric. To make sure that the greatest common divisor is unique, we assume that the greatest common divisor is monic.

Definition

Let $f(x)$ and $g(x)$ be two polynomials, not both zero. A polynomial $d(x)$ is called a greatest common divisor of $f(x)$ and $g(x)$ if

1. $d(x)$ is a common divisor of $f(x)$ and $g(x)$.
2. If $e(x)$ is a common divisor of $f(x)$ and $g(x)$, then $\mathrm{deg}(e(x))\le \mathrm{deg}(d(x))$.
3. $d(x)$ is monic.

Bézout’s Theorem

The Bézout’s theorem for polynomials is similar to the one for integers.

Theorem

Let $f(x)$ and $g(x)$ be two polynomials, not both zero. Then there exist polynomials $u(x)$ and $v(x)$ such that

$$u(x)f(x)+v(x)g(x)=\gcd (f(x),g(x)).$$

The proof of the theorem is similar to the one for integers, we will not repeat it here.

Euclidean Algorithm

The Euclidean algorithm for polynomials is similar to the one for integers. We will state the algorithm and give an example.

Theorem

Let $f(x)$ and $g(x)$ be two polynomials, not both zero. Then the greatest common divisor of $f(x)$ and $g(x)$ can be found by the following process:

$$\begin{array}{rl}f(x)& =q_{1}g(x)+r_1,\mathrm{deg}(r_1)<\mathrm{deg}(g(x)),\\ g(x)& =q_2{r}_1+r_2,\mathrm{deg}(r_2)<\mathrm{deg}(r_1),\\ r_1& =q_3{r}_2+r_3,\mathrm{deg}(r_3)<\mathrm{deg}(r_2),\\ & ⋮\\ r_{n-2}& =q_n{r}_{n-1}+r_n,\mathrm{deg}(r_n)<\mathrm{deg}(r_{n-1}),\\ r_{n-1}& =q_{n+1}{r}_n.\end{array}$$

The greatest common divisor of $f(x)$ and $g(x)$ is the monic version of $r_n$.

We will give an example in $\mathbb{Q}[x]$.

Step	Dividend	Divisor	Division Result	Remainder
1	$f(x)=x^3-x^2+2x-2$	$g(x)=x^2+2x-3$	$f=(x-3)g+(11x-11)$	$r_1(x)=11x-11$
2	$g(x)=x^2+2x-3$	$r_1(x)=11x-11$	$g=\left(\frac{x+3}{11}\right)r_1+0$	0 (stop)

Since we make the monic assumption, the greatest common divisor is $r_1(x)/11=x-1$.

In the above example, one can also observe the factorizations in $\mathbb{Q}[x]$:

$$f(x)=(x-1)(x^2+2)\text{and}g(x)=(x-1)(x+3).$$

We observe the common factor $(x-1)$ in both $f(x)$ and $g(x)$ by performing the factorization as well. We will introduce the factorization for $F[x]$ which is the analogue of the factorization for integers.