The Ring of Polynomials with Real Coefficients

Recall from the Rings page that if $+$ and $*$ are binary operations on the set $R$, then $R$ is called a ring under $+$ and $*$ denoted $(R, +, *)$ when the following are satisfied:

• 1. For all $a, b \in R$ we have that $(a + b \in R)$ (Closure under $+$).

• 2. For all $a, b, c \in R$, $a + (b + c) = (a + b) + c$ (Associativity of elements in $R$ under $+$).

• 3. There exists an $0 \in R$ such that for all $a \in R$ we have that $a + 0 = a$ and $0 + a = a$ (The existence of an identity element $0$ of $R$ under $+$).

• 4. For all $a \in R$ there exists a $-a \in R$ such that $a + (-a) = 0$ and $(-a) + a = 0$ (The existence of inverses for each element in $R$ under $+$).

• 5. For all $a, b \in R$ we have that $a + b = b + a$ (Commutativity of elements in $R$ under $+$).

• 6. For all $a, b \in R$ we have that $a * b = b * a$ (Closure under $*$).

• 7. For all $a, b, c \in R$, $a * (b * c) = (a * b) * c$ (Associativity of elements in $R$ under $*$).

• 8. There exists a $1 \in R$ such that for all $a \in R$ we have that $a * 1 = a$ and $1 * a = a$ (The existence of an identity element $1$ of $R$ under $*$).

• 9. For all $a, b, c \in R$ we have that $a * (b + c) = (a * b) + (b * c)$ and $(a + b) * c = (a * c) + (b * c)$ (Distributivity of $*$ over $+$).

We will now look at the ring of polynomials with real coefficients.

The Ring of Single-Variable Real-Valued Polynomials with Real Coefficients

We will denote the set of single-variable real-valued polynomials with real coefficients as $\mathbb{R}[x]$. For each polynomial $f, g \in \mathbb{R}[x]$ we define addition $+$ by $(f + g)(x) = f(x) + g(x)$ and we define multiplication $*$ by $(f * g)(x) = f(x)g(x)$. Let's verify all of the ring axioms.

Let $f, g, h \in \mathbb{R}[x]$. Then if $n$ is the largest degree between these polynomials then we can write $f(x) = \sum_{i=0}^{n} a_ix^i$, $g(x) = \sum_{i=0}^{n} b_ix^i$ and $h(x) = \sum_{i=0}^{n} c_ix^i$.

We have that $\mathbb{R}[x]$ is closed under $+$ since:

(1)

\begin{align} \quad f(x) + g(x) = \sum_{i=0}^{n} a_ix^i + \sum_{i=0}^{n} b_ix^i = \sum_{i=0}^{n} (a_i + b_i)x^i \in \mathbb{R}[x] \end{align}

$+$ is also associative since:

(2)

\begin{align} \quad f(x) + (g(x) + h(x)) = \sum_{i=0}^{n} a_ix^i + \left ( \sum_{i=0}^{n} (b_i + c_i) x^i\right ) = \sum_{i=0}^{n} a_ix^i + \left ( \sum_{i=0}^{n} b_ix^i + \sum_{i=0}^{n} c_ix^i \right ) \\ \quad = \left ( \sum_{i=0}^{n} a_ix^i + \sum_{i=0}^{n} b_ix^i \right ) + \sum_{i=0}^{n} c_ix^i = \left ( \sum_{i=0}^{n} (a_i + b_i)x^i \right ) + \sum_{i=0}^{n} c_ix^i = (f(x) + g(x)) + h(x) \end{align}

The identity element is the zero polynomial $z(x) = 0 \in \mathbb{R}[x]$ where:

(3)

\begin{align} \quad f(x) + z(x) = f(x) + 0 = \sum_{i=0}^{n} a_ix^i + \sum_{i=0}^{n} 0x^i = \sum_{i=0}^{n} (a_i + 0)x^i = \sum_{i=0}^{n} a_ix^i = f(x) \end{align}

(4)

\begin{align} \quad z(x) + f(x) = 0 + f(x) = \sum_{i=0}^{n} 0x^i + \sum_{i=0}^{n} a_ix^i = \sum_{i=0}^{n} (0+a_i)x^i = \sum_{i=0}^{n} a_ix^i = f(x) \end{align}

For each polynomial $f(x) \in \mathbb{R}[x]$ we have that $-f(x) \in \mathbb{R}[x]$ is the additive inverse of $f(x)$ since:

(5)

\begin{align} \quad f(x) + (-f(x)) = \sum_{i=0}^{n} a_ix^i + \sum_{i=0}^{n} -a_ix^i = \sum_{i=0}^{n} (a_i - a_i)x^i = \sum_{i=0}^{n} 0x^i = 0 = z(x) \end{align}

(6)

\begin{align} \quad (-f(x)) + f(x) = \sum_{i=0}^{n} -a_ix^i + \sum_{i=0}^{n} a_ix^i = \sum_{i=0}^{n} (-a_i + a_i)x^i = \sum_{i=0}^{n} 0x^i = 0 = z(x) \end{align}

$+$ is commutative since:

(7)

\begin{align} \quad f(x) + g(x) = \sum_{i=0}^{n} a_ix^i + \sum_{i=0}^{n} b_ix^i = \sum_{i=0}^{n} (a_i + b_i)x^i = \sum_{i=0}^{n} (b_i + a_i)x^i = \sum_{i=0}^{n} b_ix^i + \sum_{i=0}^{n} a_ix^i = g(x) + f(x) \end{align}

We have that $\mathbb{R}[x]$ is closed under $*$ since if we let $c_k = \sum_{k=0}^{n} a_lb_{k-l}$ then:

(8)

\begin{align} \quad f(x) * g(x) = \sum_{i=0}^{n} a_ix^i * \sum_{i=0}^{n} b_ix^i = \sum_{i=0}^{n} c_ix^i \in \mathbb{R}[x] \end{align}

Furthermore $*$ is associative. We will now show this through series but the reader is encouraged to verify this.

The identity polynomial for multiplication is the polynomial $1(x) = 1$.

We lastly note that multiplication $*$ is indeed distributive over $+$. Once again, this is rather tedious to write out but the reader is advised to attempt to.

The Ring of Two-Variable Real-Valued Polynomials with Real Coefficients

Let $\mathbb{R}[x, y]$ be the set of two-variable real-valued functions with real coefficients. It can be shown in a similar manner that $(\mathbb{R}[x, y], +, *)$ is a ring where $+$ is polynomial addition and $*$ is polynomial multiplication.