Question

Show that d : Q[x]\{0} -> N U {0} : d(f) = 5^deg f is a Euclidean valuation on Q[x]

Answer 1

We know that for every $A,B\in\mathbb Q[x]$ there are $P,R\in\mathbb Q[x]$ such that $A=BP+R$, where $\deg R<\deg B$ or $R=0$. Then $R=0$ or $d(R)=5^{\deg R}<5^{\deg B}=d(B)$.

We obtain that for every $A,B\in\mathbb Q[x]$ there are $P,R\in\mathbb Q[x]$ such that $A=BP+R$, where $d(R)<d(B)$ or $R=0$.