Polynomials over a Field

# Polynomials over a Field

Recall that a polynomial is an equation of the form:

(1)
\begin{align} \quad f(x) = a_0 + a_1x + ... + a_nx^n \end{align}

Given a field $(F, +, \cdot)$ we can restrict the coefficients $a_0, a_1, ..., a_n$ to be contained in $F$ and then study various properties of such polynomials that we are already familiar with (such as roots, factorability, etc…). We first formally define a polynomial over a field.

 Definition: Let $(F, +, \cdot)$ be a field. A Polynomial over $F$ is an equation of the form $f(x) = a_0 + a_1x + ... + a_nx^n$ where the coefficients $a_0, a_1, ..., a_n \in F$. The set of all polynomials over a field $F$ is denoted $F[x]$

For example, consider the field $(\mathbb{C}, +, \cdot)$ of complex numbers. An example of a polynomial over $\mathbb{C}$ is:

(2)
\begin{align} \quad f(x) = (2 + i) + (3 + i)x + 4x^2 + 2ix^3 \end{align}
 Definition: If $(F, +, \cdot)$ is a field and $f \in F[x]$ is given by $f(x) = a_0 + a_1x + ... + a_nx^n$ with $a_n \neq 0$ then the Degree of $f$ is $n$ and is denoted $\deg (f) = n$. By convention we denote $\deg (0) = -\infty$.

In the above example, $\deg (f) = 3$.

 Definition: If $(F, +, \cdot)$ is a field and $f \in F[x]$, $f(x) \neq 0$ with $f(x) = a_0 + a_1x + ... + a_nx^n$ then $f$ is said to be a **Monic Polynomial if $a_n = 1$.

For example, in the field of real numbers, an example of a monic polynomial over $\mathbb{R}$ is:

(3)
\begin{align} \quad g(x) = 1 + 2x + x^2 \end{align}

An example of a polynomial over $\mathbb{R}$ that is not monic is:

(4)
\begin{align} \quad h(x) = 1 + 2x + 3x^2 \end{align}

Given any field $(F, +, \cdot)$, we can define a new field on $F[x]$, the set of polynomials over $F$ with the binary operations of function addition and function multiplication. This is outlined in the following theorem and is easy (but tedious) to verify.

 Theorem 1: If $F$ is a field then $(F[x], +, \cdot)$ is a field where for all $f, g \in F[x]$ with $f(x) = a_0 + a_1x + ... + a_nx^n$ and $g(x) = b_0 + b_1x + ... + b_mx^m$ we define $f + g = [a_0 + a_1x + ... + a_nx^n] + [b_0 + b_1x + ... + b_mx^m]$ and $f \cdot g = [a_0 + a_1x + ... + a_nx^n][b_0 + b_1x + ... + b_mx^m]$. The additive identity in $F[x]$ is the polynomial $0(x) = 0$, and the multiplicative identity in $F[x]$ is the polynomial $1(x) = 1$.