# Proving that $\mathbb{K}$ is a skew field (see post for detailed description of $\mathbb{K}$).

Consider a Desarguesian affine space $$\mathcal{A}$$. Choose a fixed point $$o$$ and a fixed line $$\mathbb{K}$$ through $$o$$. Select another, arbitrary point $$e$$ on $$\mathbb{K}$$. For each point $$x \in \mathbb{K}$$, we write $$\vec{x}$$ instead of $$\vec{ox}$$.

Define the sum in $$\mathbb{K}$$ as follows: $$\forall a,b\in \mathbb{K}:$$ $$a+b=c$$ if $$\vec{a}+\vec{b}=\vec{c}$$, which is equivalent with $$\tau_{\vec{a}} \tau_{\vec{b}} = \tau_{\vec{c}}$$ ($$\tau$$ is a translation along the specified vector, also note the exponential notation!).

Define the product in $$\mathbb{K}$$ as follows: $$o \cdot a = o = a \cdot o$$, for all $$a\in \mathbb{K}$$ and $$a \cdot b = c,$$ $$o \notin \{a,b\},$$ if $$h_{e,b}^{-1} \tau_{\vec{a}} h_{e,b} = \tau_{\vec{c}}$$ ($$h_{e,b}$$ is the homothety with center $$o$$ that maps $$e$$ to $$b$$). We can also say that $$a \cdot b = c \iff h_{e,a}h_{e,b} = h_{e,c}$$.

Now the statement is

With the notation used above, $$\mathbb{K},+,\cdot$$ is a skew field.

Proof:

$$\mathbb{K},+$$ is a commutative group with identity $$o$$ (because the group of translation in $$\mathcal{A}$$ is commutative and the definition of the sum of elements in $$\mathbb{K}$$ is totally based on translations along vectors).

$$\mathbb{K^*},\cdot$$ is a group (follows from the last sentence of the given description, which states that multiplication of points on $$\mathbb{K}$$ is equivalent with a composition of homotheties - which also form a group, when considering homotheties with the same center).

We want to proof that $$(a+a')\cdot b = a\cdot b + a'\cdot b$$. The left term is determined by the translation $$h_{e,b}^{-1}\tau_{\vec{a}+\vec{a}'}h_{e,b}=h_{e,b}^{-1}\tau_{\vec{a}} \tau_{\vec{a}'}h_{e,b} = h_{e,b}^{-1}\tau_{\vec{a}} h_{e,b}h_{e,b}^{-1}\tau_{\vec{a}'}h_{e,b} = \tau_{\vec{a\cdot b}}\tau_{\vec{a'\cdot b}}.$$ Which proofs what we needed.

How do I proof that $$a\cdot (b+b') = a\cdot b + a \cdot b'$$? I'll have to introduce a new definition for the summation of homotheties with center $$o$$. Then I must prove that this definition corresponds to summation in $$\mathbb{K}$$. How do I find a new definition?

Or maybe somebody can give another way to prove the last part.

Thanks.