Simple and Multiple Roots of Polynomials

# Simple and Multiple Roots of Polynomials

Let $K$ be a field and let $f(x) \in K[x]$. If $F$ is a splitting field of $f(x)$ over $K$ then $f(x)$ can be split into a product of linear factors:

(1)\begin{align} \quad f(x) = (x - r_1)^{m_1}(x - r_2)^{m_2}...(x - r_n)^{m_n} \end{align}

Where $r_1, r_2, ..., r_n \in F$ are the roots of $f(x)$. We will now distinguish between different types of roots.

Definition: Let $K$ be a field and let $f(x) \in K[x]$. Let $F$ be a splitting field of $f(x)$ over $K$ so that $f(x) = (x - r_1)^{m_1}(x - r_2)^{m_2}...(x - r_n)^{m_n}$ where $r_1, r_2, ..., r_n \in F$ are the roots of $f(x)$. The Multiplicity of the root $r_i$ is $m_i$. A root $r_i$ of $f(x)$ is said to be a Simple Root if $m_i = 1$, that is, $r_i$ has multiplicity $1$. A root $r_i$ of $f(x)$ is said to be a Multiple Root if $m_i \geq 2$. |

For example, consider the following polynomial:

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

The roots of $f(x)$ are $1$, $4$, and $-2$. The multiplicities of these roots are $2$, $3$, and $1$ respectively. The root $-2$ is a simple root, and the roots $1$ and $4$ are multiple roots.