Exercise 7, § 9

Let E (resp. E') be an affine space of finite dimension $n \geq 2$ over field K with at least 3 elements (resp. over a field K') and let u be an injective mapping of E into E' mapping any three points in a straight line in E to three points in a straight line and such that the affine linear variety generated by $u(E)$ in E' is equal to E'.

(a) Show that u maps every system of affinely independent points of E to a system of affinely independent points (use Exercise 5).

(b) Let D_1, D_2 be two parallel lines in E and D'_1, D'_2 the lines of E' containing respectively $u(D_1)$ and $u(D_2)$. Show that D'_1 and D'_2 are in the same plane; if further u is surjective, show that D'_1 and D'_2 are parallel (in the contrary case, show that there would be 3 points not in a straight line in E whose images under u would be in a straight line) (cf. Exercise 17).

(c) Suppose henceforth that if D_1 and D_2 are parallel lines in E, the lines of E' containing respectively $u(D_1)$ and $u(D_2)$ are parallel. Show that if an origin a in E and the origin $a' = u(a)$ in E' are taken, there exists an isomorphism $\sigma$ of K onto a subfield K_1 of K' such that if E is considered as a vector space over K and E' as a vector space over K_1, u is an injective semi-linear mapping (relative to $\sigma$) of E into E' (§ 1, no. 13). (Consider first the case $n = 2$; given a basis $(e_1, e_2)$ of E, show that for any two elements $\alpha, \beta$ of K, the points $(\alpha + \beta)e_1$ and $(\alpha\beta)e_1$ of E can be constructed starting with the points 0, $e_1, e_2, \alpha e_1, \beta e_1$ by constructing parallels to given lines and intersections of given lines; deduce that $u(\lambda e_1) = \lambda^\sigma u(e_1)$, where $\sigma$ is an isomorphism of K onto a subfield of K', then show that also $u(\lambda e_2) = \lambda^\sigma u(e_2)$ by considering the line joining the points $\lambda e_1$ and $\lambda e_2$. Pass finally from here to the case where n is arbitrary, arguing by induction on n.) If u is bijective, show that $K_1 = K'$.

(d) Extend the result of (c) to the case where K is a field with 2 elements assuming also that u maps every system of affinely independent points of E to a system of affinely independent points of E'.