Question

Let F be a field. Prove that F is a Euclidean domain with the function $\mathit{\delta }$given by $\mathit{\delta }\mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{=}\mathbf{0}$for each nonzero $\mathit{a}\mathbf{\in }\mathit{F}$.

Short answer

It has been proved that F is a Euclidean domain with the function $\delta$given by $\delta \left(a\right)=0$for each non-zero $a\in F$.

Step-by-step solution

Given that

The function $\delta$is given by $\delta \left(a\right)=0$.

To prove that Fis a Euclidean domain the definition must be checked.

Check the properties

It is clear that $\delta \left(ab\right)\le \delta \left(a\right)\delta \left(b\right)$.

For second property, let a,b $\in F$and $b\ne 0$.

Then, $q=a{b}^{-1}\in F$and $a=bq+r$, where$r=0$.

Conclusion

Thus, it can be concluded that F is a Euclidean Domain.