Polynomial ring/Field/One variable/Euclidean division/Fact

Euclidean division (polynomial ring)

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Let ${}P,T\in K[X]$ be polynomials with ${}T\neq 0$.

Then there exist unique polynomials

${}Q,R\in K[X]$ such that

$P=TQ+R{\text{ and with }}\operatorname {deg} \,(R)<\operatorname {deg} \,(T){\text{ or }}R=0.$

Proof, Write another proof