Definition 1

— With the above notation we say that $a$ is of order $h$, or multiplicity $h$ relative to $f$. If $h > 0$ we also say that $a$ is a root of order $h$ or multiplicity $h$ off. A root of order 1 is called a simple root, a root of order 2 a double root,... A root of order $> 1$ is said to be multiple.