Exercise 7, § 9

Let E (resp. E’) be a finite-dimensional affine space $n \geq 2$ over a field K having at least 3 elements (resp. over a field K’), and let $u$ be an injective mapping of E into E’ transforming any three points on a straight line of E into three points on a straight line, and such that the affine linear variety generated by $u(E)$ in E’ is equal to E’. (a) Now prove that $u$ transforms every system of affinely independent points of E into a system of affinely independent points (use Exercise 5 of II, p. 200). (b) Let $D_1, D_2$ be two parallel lines in E, $D'_1, D'_2$ the lines of E’ containing respectively $u(D_1)$ and $u(D_2)$. Now prove that $D'_1$ and $D'_2$ are in the same plane; if moreover $u$ is surjective, now prove that $D'_1$ and $D'_2$ are parallel (in the contrary case, now prove that there would be 3 points not on a straight line of E whose images by $u$ would be on a straight line) (cf. II, p. 204, Exercise 17). (c) Suppose henceforth that if $D_1$ and $D_2$ are parallel lines of E, the lines of E’ containing respectively $u(D_1)$ and $u(D_2)$ are parallel. Now prove that if one takes in E an origin $a$ and in E’ the origin $a' = u(a)$, 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 a semilinear injective mapping (relative to $\sigma$) of E into E’ (II, p. 32). (First consider the case $n = 2$; given a basis $(e_1, e_2)$ of E, now prove that for any two elements $\alpha, \beta$ of K, one can construct the points $(\alpha + \beta)e_1$ and $(\alpha\beta)e_1$ of E from the points $0, e_1, e_2, \alpha e_1, \beta e_1$ by constructions of parallels to given lines and intersections of given lines; deduce that one can write $u(\lambda e_1) = \lambda^\sigma u(e_1)$, where $\sigma$ is an isomorphism of K onto a subfield of K’, then show that one also has $u(\lambda e_2) = \lambda^\sigma u(e_2)$ by considering the line joining the points $\lambda e_1$ and $\lambda e_2$. Finally pass from there to the case where $n$ is arbitrary, by reasoning by induction on $n$.) If $u$ is bijective, now prove that $K_1 = K'$. (d) Extend the result of c) to the case where K is a field with 2 elements by supposing in addition that $u$ transforms every system of affinely independent points of E into a system of affinely independent points of E’.