Constructible numers are an algebraic field which can be obtanined by a finite number of algebraic extensions of degree $2^k$ from $\mathbb{Q}$. I believe this is equivalent to define the set of constructible numbers $\mathbb{K}$ as:
$$\mathbb{K} = \left\{x\in \mathbb{R}|\ \exists q_0\dots \exists q_n:x=q_0\pm\sqrt{q_1 \pm \sqrt{\dots \pm \sqrt{q_n}}}\text{, with }q_i\in\mathbb{Q} \right\}$$
My question is: if $\mathbb{K}^n = \mathbb{K}\times \dots \times\mathbb{K}$ constitue valid models for the axioms of Euclidean Geometry (since any construction with ruler and compass, starting from a segment of length 1, gives rise to points on an n-dimensional space whose coordinates would belong to $\mathbb{K}$)? [I understand that $\mathbb{K}^n$, being a countable set, is not suitable for real analysis]
Additional comment: I found, that basic Euclidean Geometry can be developed over any Pythagorean field $\mathbb{P}$, an ordered field for which:
$$p,q\in \mathbb{P} \Rightarrow \sqrt{p^2+q^2}\in \mathbb{P}$$
the smallest such a Pythagorean field is called ''Hilber field'' a countable ordered field.