A discrete form of the theorem that each field endomorphism of $\mathbb{R}{\left( {\mathbb{Q}_{{\text{p}}} } \right)}$ is the identity

Summary.

Let K be a field and F denote the prime field in K. Let $\tilde{K}$ denote the set of all r ∈K for which there exists a finite set A(r) with ${\left\{ r \right\}} \subseteq A{\left( r \right)} \subseteq K$ such that each mapping $f:A(r) \to K$ that satisfies: if 0 ∈A(r), then f(0)=0, if 1 ∈A(r), then f(1)=1, if a, b ∈A(r) and a+b ∈A(r), then f(a+b)=f(a)+f(b), if a, b ∈A(r) and a·b ∈A(r), then f(a·b)=f(a)·f(b), satisfies also f(r)=r. Obviously, each field endomorphism of K is the identity on $\tilde{K}$. We prove: $\tilde{K}$ is a countable subfield of K; if char(K)≠ 0, then $\tilde{K}$ = F; $\tilde{\mathbb{C}}=\mathbb{Q}$, $\tilde{\mathbb{R}}$ is equal to the field of real algebraic numbers, $\tilde{\mathbb{Q}_p}$ is equal to the field $\begin{aligned} & {\left\{ {x \in \mathbb{Q}_{{\text{p}}} :x\,{\text{is}}\,{\text{algebraic}}\,{\text{over}}\mathbb{Q}} \right\}} \\ & \\ \end{aligned}$.