Question

Let R be a Euclidean domain. If the function$\mathit{\delta }$ is a constant function, prove that R is a field.

Short answer

It is proved that R is a field

Step-by-step solution

Definition of Associate

An integral domain R is said to be Euclidean domain if there is a function $\mathbf{\delta }$ from the non-zero elements of R to non-negative integers with these properties:

1. If$\mathbf{a}\mathbf{}\mathbf{and}\mathbf{}\mathbf{b}$ are non-negative elements of R, then $\mathit{\delta }\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{\le }\mathit{\delta }\mathbf{\left(}\mathbf{a}\mathbf{b}\mathbf{\right)}$.
2. If $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{R}$and$\mathit{b}\mathbf{\ne }{\mathbf{0}}_{\mathbf{R}}$ , then there exist$\mathit{q}\mathbf{,}\mathit{r}\mathbf{\in }\mathit{R}$ such thatrole="math" localid="1654664477403" $\mathit{a}\mathbf{=}\mathit{b}\mathit{q}\mathbf{+}\mathit{r}$ and either$\mathit{r}\mathbf{=}{\mathbf{0}}_{\mathbf{R}}$ or$\mathit{\delta }\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{\le }\mathit{\delta }\mathbf{\left(}\mathbf{b}\mathbf{\right)}$.

Now, letrole="math" localid="1654664604902" $\delta$ be constant function defined as $\delta \left(a\right)=a$.

Use theorem 10.2

Which is says that Let R be a Euclidean domain and u a nonzero element of R. Then following are Equivalent:

1. uis a unit
2. $\mathit{\delta }\mathbf{\left(}\mathbf{u}\mathbf{\right)}\mathbf{=}\mathit{\delta }\mathbf{\left(}{\mathbf{1}}_{\mathbf{R}}\mathbf{\right)}$
3. $\mathit{\delta }\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{=}\mathit{\delta }\mathbf{\left(}\mathbf{u}\mathbf{c}\mathbf{\right)}$for some nonzero $\mathit{c}\mathbf{\in }\mathit{R}$

Therefore, by (iii)$\delta \left(c\right)=\delta \left(uc\right)\Rightarrow c=uc\Rightarrow u=1$ .

Now, It is true for all $a\in R$

Therefore all$a\in R$ are unit element

Hence $a^{-1}$exist for all $a\in R$

Hence$R$ is a field.