Polynomial ring/Field/1/Common divisor/Section

Definition

Let ${}P_{1},\ldots ,P_{n}\in K[X]$ denote polynomials over a field ${}K$. We say that a polynomial ${}T\in K[X]$ is a common divisor of the given polynomials if ${}T$ divides

every ${}P_{i}$.

Definition

Let ${}P_{1},\ldots ,P_{n}\in K[X]$ denote polynomials over a field ${}K$. We say that a polynomial ${}G\in K[X]$ is a greatest common divisor of the given polynomials, if ${}G$ is a common divisor of the ${}P_{i}$, and if ${}G$ has among all common divisors of the ${}P_{i}$ maximal

degree.

A greatest common divisor is not uniquely determined, since with ${}G$ also ${}cG$ for some constant ${}c\neq 0$ is also a greatest common divisor. However, if we restrict to normed polynomials, then the greatest common divisor is unique.

Definition

Polynomials ${}P_{1},\ldots ,P_{n}\in K[X]$ over a field ${}K$ are called coprime, if they have, with the exception of constants ${}c\neq 0$, no

common divisor.