As far as I can judge, an interesting question "Why are fields defined the way they are?" was asked recently. Unfortunately, it was not specific enough. Which is still, in my opinion, intrinsically linked to the problem raised.
Personally, I would have answered that we can do geometry on a skew field like $\mathbb H$, which is not a field. It could appear a free and excessively abstract game for beginners. But we know that this is anything but the case, because for example, Pappus theorem in a projective plane, where the skew field is $\mathbb H$, is false. Besides, I would have liked to redo the demonstration in a possible response.
Are there any other insightful examples that show the interest of vector space over a skew field in geometry?
