Proposition 12

*Let A be a maximal ordered field, E an oriented plane over A, $D_0$ a half-line of E, and G the set of half-lines of E distinct from $D_0$. The totally ordered sets A and G (prop. 11) are isomorphic.

Let, indeed, $(x, y)$ be a basis of E such that $x \in - D_0$ and the bivector $x \wedge y$ be $> 0$. To every element $t$ of A let us correspond the half-line $f(t)$ to which the vector $(1 - t^2)x + 2ty$. It is clear that $f(A) \subset G$. Let us prove that $f$ is strictly increasing. Indeed, in order that the sequence $D_0, f(t), f(t')$ ($t, t'$ in A) be direct, it is necessary and sufficient, by definition, that at least two of the elements

let them be $> 0$. Now the second is equal to $2(t' - t)(1 + tt')$. Hence, if $t < t'$, then either $tt' \geqslant 0$, whence $t' > 0$ or $-t > 0$, or $tt' < 0$, whence $-t > 0$ and $t' > 0$; in any case $D_0, f(t), f(t')$ is direct. Since $A$ is totally ordered, $f$ is an isomorphism of $A$ onto the ordered set $f(A)$ (Ens, Chapter III, § 1, No. 14, Prop. 13).

It remains to prove that $f$ is surjective. For this, consider the positive form $\Phi$ on $E$ such that $(x, y)$ is an orthonormal basis for $\Phi$. For every half-line $D \in G$, there exists an angle $\varphi$ and only one such that $2\varphi = (\widehat{-D_0, D})$ (No. 1, Prop. 3); since $(\widehat{-D_0, D})$ is not the straight angle, $\varphi$ is not the right angle, and $\operatorname{tg} \varphi$ is therefore finite. Then, by virtue of formulas (16) (No. 3), we have $D = f(\operatorname{tg} \varphi)$. This completes the proof.

Exercises. 1) With the notation of No. 1, put $Q(x) = \Phi(x, x)$; the vector space $E$ is canonically identified with $C^-(Q)$ (§ 9, No. 1). For every $z \in C^+(Q)$, and every $x \in E$, we have $zx \in E$; prove that $x \to zx$ is an element of $A(\Phi)$, and that $z \to s_z$ is an isomorphism of $C^+(Q)$ onto the algebra $A(\Phi)$.

2) The hypotheses and notation are those of No. 1 and of Exercise 1.

a) Let $C$ be the set of $x \in E$ such that $\Phi(x, x) = 1$ (unit circle), and let $\mathscr{D}$ be the set of lines $D$ whose intersection with $C$ is not empty (and consequently consists of two opposite elements of $E$). A pointed line is, by definition, any pair $\Delta = (D, z)$ formed of a line $D \in \mathscr{D}$ and one of the points $z \in D \cap C$. Now prove that if $\Delta_1 = (D_1, z_1)$, $\Delta_2 = (D_2, z_2)$ are two pointed lines, there exists one and only one rotation $u$ such that $u(z_1) = z_2$ (and consequently $u(D_1) = D_2$), which is expressed by writing $u(\Delta_1) = \Delta_2$. In the set of pairs $(\Delta_1, \Delta_2)$ of pointed lines, the relation “there exists a rotation $u$ such that $u(\Delta_1) = \Delta_1'$ and $u(\Delta_2) = \Delta_2'$” is an equivalence relation. The set $\mathfrak{U}_1$ of equivalence classes of pairs of pointed lines with respect to this relation is called the set of angles of pointed lines, and the equivalence class to which a pair $(\Delta_1, \Delta_2)$ of pointed lines belongs is called the angle of this pair and denoted by $(\widehat{\Delta_1, \Delta_2})$; the relation $(\widehat{\Delta_1, \Delta_2}) = (\widehat{\Delta_1', \Delta_2'})$ is equivalent to $(\widehat{\Delta_1, \Delta_1'}) = (\widehat{\Delta_2, \Delta_2'})$ and the rotation which transforms $\Delta_1$ into $\Delta_2$ is said to be a rotation of angle $0 = (\widehat{\Delta_1, \Delta_2})$ and is denoted by $h_1(\theta)$; $h_1$ is a bijection of $\mathfrak{U}_1$ onto $O^+$, and the commutative group structure of $O^+$ is transported to $\mathfrak{U}_1$ by means of $h_1^{-1}$, the group $\mathfrak{U}_1$ thus defined being written additively; the angle of the pair formed by a pointed line $(D, z)$ and the “opposite” pointed line $(D, -z)$, corresponding to the rotation $x \to -x$, is again called the flat angle in $\mathfrak{U}_1$.

b) In the set $\mathscr{D}_0 \supset \mathscr{D}$ of non-isotropic lines, one defines as in No. 3 the notion of angle of lines, the group $\mathfrak{U}_0$ (using Corollary 3 of Proposition 1 of No. 1), the right angle in $\mathfrak{U}_0$, and the canonical bijection $h$ of $\mathfrak{U}_0$ onto $S^+/H$. With the notation of the corollary of Proposition 3, put $\bar{d} = h_1^{-1} \circ d \circ h$ and $\bar{i} = h^{-1} \circ i \circ h_1$, so that $\bar{i}$ is a homomorphism of $\mathfrak{U}_1$ into $\mathfrak{U}_0$ and $\bar{d}$ a homomorphism of $\mathfrak{U}_0$ into $\mathfrak{U}_1$; $\bar{d}$ is bijective, and the kernel of $\bar{i}$ consists of '0 and the flat angle; moreover one has $\bar{d}(\bar{i}(\theta)) = 2\theta$ for $\theta \in \mathfrak{U}_1$ and $\bar{i}(\bar{d}(\varphi)) = 2\varphi$ for $\varphi \in \mathfrak{U}_0$. In order that $\bar{i}$ be surjective (in other words, in order that $\mathscr{D}$ be the set of all non-isotropic lines), it is necessary and sufficient that the field A be Pythagorean (Chapter VI, § 2, Exercise 8 d)) and that there exist an orthonormal basis for $\Phi$; this comes to the same as saying that $\Phi(x, x)$ is a square for every $x \in E$.

3) The hypotheses and notation are those of Exercise 2. a) Let s be a symmetry with respect to a non-isotropic line D ($§ 6$, No. 4). For every pointed line $\Delta_1 = (D_1, z_1)$, let

and let $\varphi = (\widehat{D_1}, \widehat{D})$; now prove that one has $(\widehat{\Delta_1}, \widehat{\Delta_2}) = \bar{d}(\varphi)$. b) Now prove that every orthogonal transform of determinant -1 is a symmetry s with respect to a non-isotropic line (cf. § 6, Exercise 15 e)); for every rotation u, one has $sus^{-1} = u^{-1}$. c) If $x, y$ are any two points of E such that $\Phi(x, x) = \Phi(y, y) \neq 0$, there exists one and only one symmetry with respect to a non-isotropic line, which transforms $x$ into $y$. d) Let s, s' be the symmetries with respect to two non-isotropic lines D, D', and let $\varphi = (\widehat{D}, \widehat{D'})$; in order that s’s be a rotation of angle $\theta$, it is necessary and sufficient that $\bar{d}(\varphi) = \theta$. e) Now prove that the commutator subgroup of the orthogonal group O(Q) is the image of $\mathfrak{U}_1$ under the homomorphism $\theta \to h_1(2\theta)$ ($§ 6$, Exercise 17 a)); in order that this image be equal to $O^+$, it is necessary and sufficient that the homomorphism $\bar{i}$ be surjective (Exercise 2 b)).

4) The hypotheses and notations are those of Exerc. 2. Let $a, b$ be two points of the circle C, $\Delta_a, \Delta_b$ the pointed lines passing through $a$ and $b$ (and through 0) respectively, and let $\theta = (\widehat{\Delta_a}, \widehat{\Delta_b})$. For every $x \in E$, distinct from $a$ and $b$, let $D_{xa}$ (resp. $D_{xb}$) be the affine line passing through $a$ and $x$ (resp. $b$ and $x$), and let $D'_{xa}$ (resp. $D'_{xb}$) be the direction of $D_{xa}$ (resp. $D_{xb}$); show that, for $x \in C$ ($x$ distinct from $a$ and $b$), it is necessary and sufficient that the angle $\varphi = (\widehat{D'_{xa}}, \widehat{D'_{xb}})$ be such that $\bar{d}(\varphi) = \theta$ (use Exerc. 3). How is this result modified when $x = a$ or $x = b$ (cf. § 6, Exerc. 25 a))?

5) With the notations of Nos. 1 and 2 and of Exerc. 2, suppose that $\Phi$ is of index 1, in other words that $\delta$ is a square $\gamma^2$ in A; denote by $D_1, D_2$ the isotropic lines of E, containing respectively the vectors $e_1 - \frac{1}{\gamma} e_2$ and $e_1 + \frac{1}{\gamma} e_2$. a) Show that the group of rotations $O^+$ is isomorphic to the multiplicative group $A^*$ of the field A. b) For every angle $\theta \in \mathfrak{U}_1$, put $e_w(\theta) = c_w(h_1(\theta)) + \gamma s_w(h_1(0))$. Show that $\theta \to e_w(\theta)$ is an isomorphism of $\mathfrak{U}_1$ onto the group $A^*$.

c) Let D, D' be any two lines, and let $\varphi = (\widehat{D}, \widehat{D'})$. Show that the cross ratio $\begin{bmatrix} D_1 & D_2 \\ D' & D \end{bmatrix}$ (Chapter II, 2nd ed., App. III, Exerc. 5) is equal to $e_w(\bar{d}(\varphi))$ ("Laguerre formula"). (Observe that $D_1$ and $D_2$ are invariant under every direct similarity, and by using Exerc. 4 c) of Chapter II, 2nd ed., App. III, reduce to the case where $D = Ae_1$.)

6) Suppose that A is an ordered field. a) Let $D_1, D_2$ be two non-isotropic half-lines with origin 0. Show that there exists a direct similarity $u$ such that $u(D_1) = D_2$; every other direct similarity having this property is of the form $v = su$, where s is a homothety with ratio $> 0$.

b) On the set of pairs $(D_1, D_2)$ of non-isotropic half-lines, the relation “there exists a direct similitude $u$ such that $u(D_1) = D'_1$ and $u(D_2) = D'_2$” is an equivalence relation. The set $\mathfrak{A}$ of equivalence classes of non-isotropic half-lines, with respect to this relation, is called the set of angles of half-lines (non-isotropic), and the equivalence class of a pair $(D_1, D_2)$ of such half-lines is called the angle of this pair and denoted by $(\widehat{D_1}, \widehat{D_2})$; if $\theta = (\widehat{D_1}, \widehat{D_2})$, one says that $\theta$ is the angle of every direct similitude transforming $D_1$ into $D_2$; let $h_2(\theta)$ be the class mod. $H^+$ of these similitudes, so that $h_2$ is a bijection of the set $\mathfrak{A}$ onto the group $S^+/H^+$; one transports to $\mathfrak{A}$ by means of $h_2^{-1}$ the commutative group structure of $S^+/H^+$, denoting additively the group $\mathfrak{A}$ thus defined. Define a canonical injection $\bar{j}$ of the group $\mathfrak{A}_1$, of the angles of pointed lines (exerc. 2) into the group $\mathfrak{A}$, such that $h_2 \circ \bar{j} \circ h_2^{-1}$ is the canonical injection $j$ of $O^+$ into $S^+/H^+$. For $\bar{j}$ to be surjective, it is necessary and sufficient that the homomorphism $\bar{i}$ of $\mathfrak{A}_1$ into $\mathfrak{A}_0$ (exerc. 2 b)) be surjective.

c) Now prove that in $\mathfrak{A}$ the equation $2\theta = 0$ has 2 solutions if $\delta < 0$ and 4 solutions if $\delta > 0$. In the first case, the solution $\varpi \neq 0$ of this equation is again called the flat angle.

¶ 7) The hypotheses and notations being those of exerc. 6, suppose the homomorphism $\bar{j}$ bijective; one then defines cos $\theta$ and sin $\theta$ for every $\theta \in \mathfrak{A}$ as in No. 3. Let T be the set of $\theta \in \mathbf{A}$ such that $\sin \theta \geqslant 0$.

a) Now prove that for every $\theta \in T$, there exists an angle $\theta' \in T$, and only one, such that $2\theta' = \theta$; one puts $\theta' = \theta/2$.

b) Let L be the $\mathbf{Z}$-module of formal linear combinations of elements of T with coefficients in $\mathbf{Z}$ (Chapter II, § 1, No. 8); denote by $\xi + \eta$ and $-\xi$ the sum and the opposite in L. Let N be the submodule of L generated by the elements of L of the form $\xi + \eta - (\xi + \eta)$ for all pairs $(\xi, \eta)$ of elements of T such that $\xi + \eta \in T$ (sum taken in the group $\mathfrak{A}$). Let $\dot{f}$ be the homomorphism of L into $\mathfrak{A}$ which extends the canonical injection of T into $\mathfrak{A}$, and $\dot{g}$ the endomorphism of L which extends the mapping $\theta \to \theta/2$ of T into itself. We have $\dot{f}(N) = \{0\}$ and $\dot{g}(N) \subset N$; by passing to quotients, we deduce from $\dot{f}$ a homomorphism $f$ of $M = L/N$ into $\mathfrak{A}$, and from $\dot{g}$ an endomorphism $g$ of $M$; we put $g(\mu) = \mu/2$ and if $g^m$ is the m-th iterate of g, $g^m(\mu) = 2^{-m}\mu$; we have $2^m(2^{-m}\mu) = \mu$ for every $\mu \in M$.

c) Show that the restriction to T of the canonical mapping $\psi$ of L onto $M = L/N$ is injective, which makes it possible to identify T with a subset of M by means of $\psi$. Show that, if $\lambda_1, \ldots, \lambda_m$ are elements $\neq 0$ of T, the sum $\lambda_1 + \lambda_2 + \cdots + \lambda_m$ cannot be 0 in M (consider the element $2^{-m}(\lambda_1 + \cdots + \lambda_m)$ and reason by induction on m, observing that these elements belong to T).

d) Let $M_+$ be the set of finite sums (in M) of elements of T; show that $M_+ \cap (-M_+) = \{0\}$ and $M = M_+ \cup (-M_+)$, and consequently that $M_+$ is the set of elements $\geqslant 0$ for a totally ordered group structure on M (it will be noted that for every $\mu \in M_+$, there exists an integer m such that $2^{-m}\mu \in T$); this totally ordered group is called the group of measures of angles of half-lines. Show that the homomorphism $f$ of M into $\mathfrak{A}$ is surjective, and that its kernel is the set of integer multiples of $2\pi$, where $\pi$ is the straight angle (Exercise 6 c)). Prove that T is identified with the interval $[0, \pi]$ in M (establish by induction on m that if $\mu, \lambda_1, \ldots, \lambda_m$ belong to T and if one has $\lambda_1 + \cdots + \lambda_m \leqslant \mu$, then $\lambda_1 + \cdots + \lambda_m \in T$). Show that in T (thus identified with an interval of M), the function $\theta \to \cos \theta$ is strictly decreasing.

e) In order that the totally ordered group M be Archimedean (Chapter VI, § 1, Exercise 31), it is necessary and sufficient that the additive group of the field A be Archimedean. (To see that the condition is necessary, observe that if sin $\theta$ is infinitely small with respect to the subfield $\mathbf{Q}$ of A (Chapter VI, § 2, Exercise 1), the same is true of sin $n\theta$ for every integer n. To see that the condition is sufficient, observe that if $0 \leqslant \theta \leqslant \pi/4$, one has sin $2\theta \geqslant \sqrt{2} \sin \theta$).

8) Let E be an oriented plane over an ordered field A. Let D', D'' be two distinct half-lines; let $x'$ (resp. $x''$) be a vector $\neq 0$ in D' (resp. D''); the angular sector (open or closed) with origin D' and extremity D'' is said to be salient (resp. re-entrant, straight) if $x' \wedge x'' > 0$ (resp. $x' \wedge x'' < 0, x' \wedge x'' = 0$).

a) In order that there exist an automorphism of the vector space E transforming an open (resp. closed) angular sector $\Sigma_1$ into an open (resp. closed) angular sector $\Sigma_2$, it is necessary and sufficient that $\Sigma_1$ and $\Sigma_2$ should be both salient, or both re-entrant, or both flat.

b) Show that the ordered set $\{D', D''\}$ is isomorphic to the interval $[0, 1]$ of A (first consider the case of a salient sector and remark that, in A, any two bounded closed intervals are isomorphic ordered sets).

c) With the notations and hypotheses of Exercise 7, define a canonical bijective mapping of the set T onto a flat angular sector, and show that this mapping is an isomorphism for the order structures.

9) Let A be a Pythagorean ordered field, E a finite-dimensional vector space over A, Q a positive nondegenerate quadratic form on E, for which E admits an orthonormal basis (that is to say, Q(x) is a square in A for every x ∈ E). Show that the commutator group Ω(Q) of the orthogonal group O(Q) is the rotation group O⁺(Q) = SO(Q) (use Exercise 3 e) of § 10 and Exercise 17 a) of § 6).

10) Let A be a maximal ordered field, E a finite-dimensional vector space over A, Q a positive nondegenerate quadratic form on E. For every orthogonal transform u ∈ O(Q), show that there exists a decomposition of E into the direct sum of pairwise orthogonal subspaces P, N, R_i (1 ≤ i ≤ r) having the following properties: 1° u(x) = x in P, u(x) = −x in N ; 2° each of the R_i is of dimension 2, one has u(R_i) = R_i and the restriction of u to R_i is a rotation through an angle θ_i, distinct from 0 and from the flat angle. Moreover, for two decompositions of this nature, the subspaces P and N are the same, as well as the sequence of elements cos θ_i, apart from order (cf. § 7, No. 3, Corollary 2 of Theorem 2). Deduce that every rotation u ∈ O⁺(Q) is a commutator tst⁻¹s⁻¹, where s and t are in O(Q) and s² = 1 (cf. Exercise 3 d)).

11) Let A be a maximal ordered field, L the field of quaternions over A (relative to the pair (−1, −1)), E the subspace (of dimension 3) of L consisting of pure quaternions (§ 9, Exercise 15 a)) ; every quaternion is therefore written in one and only one way s = α . 1 + v, where α ∈ A and v ∈ E ; one has α² − v² = ρ . 1, where ρ ∈ A and ρ ≥ 0 ; put \|v\| = \sqrt{\rho}.

a) Let φ(s) be the rotation x → sx s⁻¹ in E, for the positive nondegenerate quadratic form x → \|x\|^2 on E (§ 9, No. 5, Theorem 4 and Exercise 15). Show that if v ≠ 0, the vectors of the line D ⊂ E containing v are invariant under φ(s) ; the restriction of φ(s) to the orthogonal plane P = D⁰ is a rotation through an angle θ such that (for a suitable orientation of P) one has tg \frac{\theta}{2} = \|v\|/α. (If (1, i, j, k) is the canonical basis of L over A, reduce to the case where v = βi, β ∈ A, and then calculate sjs⁻¹).

b) Show that every quaternion of norm 1 can be written tst⁻¹s⁻¹, where s ∈ L, t ∈ E (cf. Exercise 10), and that, if a, b are two pure quaternions of norm 1, there exists a quaternion s such that b = sas⁻¹.

c) In order that two quaternions s = α + v, t = β + w, where α, β are in A, v, w in E, should commute, it is necessary and sufficient that v = λw, λ ∈ A ; in order that st = −ts, it is necessary and sufficient that α = β = 0 and that the vectors v and w in E should be orthogonal (cf. § 3, Exercise 10).

12) Let A be a maximal ordered field, L a Euclidean space of dimension n over A, whose metric form Φ is positive nondegenerate ; let d(x, y) denote the distance \sqrt{\Phi(x - y, x - y)} of two points of L (§ 7, No. 1, Remark). For every point c ∈ L and every element ρ > 0 of A, the set of x ∈ L such that d(x, c) = ρ is called the sphere with center c and radius ρ ; the spheres are therefore nondegenerate affine quadrics in L, admitting a center (§ 6, Exercise 25).

a) Show that at every point x of a sphere S with center c, the tangent hyperplane to S at x (§ 6, Exercise 25) is perpendicular (§ 6, Exercise 22) to the line through c and x.

b) Let S be a sphere with center c and radius ρ, a a point of L, D a line through a and meeting S in two distinct points x₁, x₂ (resp. tangent to S at a point x). Show that one has

The element $(d(a, c))^2 - \rho^2$ of A is called the power of a with respect to S.

c) Let S₁, S₂ be two spheres not having the same center ; show that the set of points of L whose powers with respect to S₁ and S₂ are equal is a hyperplane perpendicular to the line through the centers of the two spheres, and containing the intersection S₁ ∩ S₂ ; one says that this hyperplane is the radical hyperplane of S₁ and S₂.

d) Let S₁, S₂ be two spheres with respective centers c₁, c₂, and respective radii ρ₁, ρ₂. Show that the following properties are equivalent :

α) The intersection S₁ ∩ S₂ is not empty and, for every point of this intersection, the tangent hyperplanes to S₁ and S₂ at this point are perpendicular.

$\beta_1$ The power of c₁ with respect to S₂ is $\rho_1^2$.

$\beta_2$ The power of c₂ with respect to S₁ is $\rho_2^2$.

$\gamma_1$ The radical hyperplane of S₁ and S₂ is the polar hyperplane (§ 6, Exercise 25) of c₁ with respect to S₂.

$\gamma_2$ The radical hyperplane of S₁ and S₂ is the polar hyperplane of c₂ with respect to S₁.

When these properties are satisfied, one says that the spheres S₁, S₂ are orthogonal.

e) Let S₁, S₂ be two orthogonal spheres, $c_1, c_2$ their centers. Show that if $\varpi_1, \varpi_2$ are the powers of a point x with respect to S₁, S₂ respectively, one has $\varpi_1 + \varpi_2 = 2\Phi(x - c_1, x - c_2)$. Converse.

13) The hypotheses and notations are the same as in Exercise 12. Given a point c ∈ L and an element $\alpha \neq 0$ of A, the involutive permutation u of the set $L - \{ c \}$ such that, for every $x \in L - \{ c \}$, $u(x)$ belongs to the line passing through c and x and satisfies the relation $\Phi(x - c, u(x) - c) = \alpha$ is called the inversion with pole c and power $\alpha$. By abuse of language, one says that u is an inversion in L.

a) If u, v are two inversions with the same pole c and with powers $\alpha, \beta$, $uv^{-1}$ is the restriction to $L - \{ c \}$ of the homothety (Chapter II, 2nd ed., App. II, Exercise 6) with center c and ratio $\alpha \beta^{-1}$.

b) Let S be a sphere (Exercise 12) containing c. Show that the image of $S - \{ c \}$ under an inversion with pole c is a hyperplane perpendicular to the line joining c to the center of S (by abuse of language, this hyperplane is said to be the image of S under the inversion considered). Converse.

c) Let S be a sphere not containing c. Show that, if $\varpi$ is the power of c with respect to S (Exercise 12 b)), the image of S under an inversion with pole c and power $\alpha$ is the image of S under a homothety with center c and ratio $\alpha / \varpi$. If $n = 2$ and if, for every $x \in S$, T (resp. T’) denotes the tangent to S (resp. $u(S)$) at the point x (resp. $u(x)$), D the line passing through c, x and $u(x)$, show that one has $(\widehat{D, T}) = (\widehat{T', D})$.

d) Let $S_1, S_2$ be two orthogonal spheres (Exercise 12 d)), $S'_1, S'_2$ their images under an inversion with pole $c$. If $c$ does not belong to $S_1$ or to $S_2$, show that $S'_1$ and $S'_2$ are orthogonal spheres. If $c \in S_1$ and $c \notin S_2$, $S'_1$ is a hyperplane containing the center of $S'_2$. If $c \in S_1 \cap S_2$, $S'_1$ and $S'_2$ are perpendicular hyperplanes (§ 6, Exercise 22). Converses.

e) Let $u$ be an inversion with pole $c$ and power $\alpha = \rho^2 > 0$ and C the sphere with center $c$ and radius $\rho$. If $x_1, x_2$ are two distinct points situated on a line passing through $c$, and distinct from $c$, the following properties are equivalent: $\alpha )$ $x_1$ and $x_2$ are transformed into one another by $u ; \beta )$ $x_1$ and $x_2$ are conjugate with respect to $C$ (§ 6, Exercise 25); $\gamma )$ every sphere containing $x_1$ and $x_2$ is orthogonal to $C$. One also says that $u$ is the inversion of sphere $C$.

¶ 14) The hypotheses and notations being the same as in Exercise 12, choose an origin 0 in L. Let $E_1$ be the vector space direct sum of L and of a space $Af_1$ of dimension 1; denote by $Q_1$ the quadratic form on $E_1$ such that for $x \in L$ and $\eta \in A$, one has

a form which is positive and non-degenerate; denote by $C$ the sphere with center 0 and radius 1 in $E_1$ (for $Q_1$).

In the Euclidean space $E_1$, let $s$ be the inversion with pole $-f_1$ and power 2 (Exercise 13); its restriction $s_0$ to L transforms L into $C - \{ -f_1 \}$; $s_0$ (resp. $s_0^{-1}$) is called, by an abuse of language, the stereographic projection of L onto C (resp. of C onto L) from the point of view $-f_1$. For every inversion $u$ in L, with pole $c$, $s_0 us_0^{-1}$ is an involutive permutation of the complement in C of the set $\{ s_0(c), -f_1 \}$; this is extended to an involutive permutation $u'$ of C by putting $u'(s_0(c)) = -f_1,\ u'(-f_1) = s_0(c)$, and one says that $u'$ is an inversion in C. Analogously, for every symmetry $\nu$ in L with respect to a hyperplane, $s_0 \nu s_0^{-1}$ is an involutive permutation of the complement in C of the set $\{ -f_1 \}$; this is extended to an involutive permutation $\nu'$ of C by putting $\nu'(-f_1) = -f_1$ and one says that $\nu'$ is a symmetry in C. The group of permutations of C generated by the inversions and symmetries is called the conformal group of C (or of L, by an abuse of language).

a) Now prove that the conformal group of C is generated by the symmetries $\nu'$ and the inversions $u'$ corresponding to inversions $u$ in L of power $> 0$. (Use Exercise 13 a) and remark that in L every translation, as well as the homothety $x \to -x$, is a product of symmetries with respect to hyperplanes).

b) Let $u$ be an inversion in L of power $> 0$; now prove that the corresponding inversion $u'$ in C is the restriction to C of a well-determined transform $u'_1$ which is either an inversion of power $> 0$ in $E_1$, whose sphere (Exercise 13 e)) is orthogonal to C, or a symmetry with respect to a hyperplane of $E_1$ passing through 0 (consider in $E_1$ the inversion $u_1$ with the same pole and the same power as $u$). Formulate the corresponding proposition for the symmetry $\nu'$ in C corresponding to a symmetry $\nu$ in L with respect to a hyperplane.

c) In the vector space $E_2 = A \times E_1$, consider the quadratic form $Q_2$ such that, for $\zeta \in A, y \in E_1$, one has

a form which is nondegenerate and of signature $(1, n + 1)$. Identify $E_1$ with its canonical image in the projective space $\mathbf{P}(E_2)$ (Chapter II, 2nd ed., App. III, No. 4). Now prove (with the notations of $b$) that $u'$ is also the restriction to $C$ of a projective linear mapping $\bar{u}''$ arising by passing to the quotients from a transform $u''$ of the orthogonal group $O(Q_2)$, which is a symmetry with respect to a nonisotropic hyperplane in $E_2$. State the corresponding proposition for $\varphi'$. Deduce that the conformal group of $L$ is isomorphic to the quotient of the group $O(Q_2)$ by its center (use proposition 5 and Exercise 17 c) of § 6). Conclude from this that every transform of the conformal group is a product of at most $n + 2$ transforms which are inversions or symmetries in $L$ (cf. § 6, Exercise 15 e)).

$d$ ) Let $\Sigma$ be the set whose elements are the spheres and the hyperplanes in the affine space $L$. Deduce from $b$ that there exists a bijection of $\Sigma$ onto the complement, in $\mathbf{P}(E_2)$, of the set of $x \in E_1$ such that $Q_1(x) \leqslant 1$, such that to two orthogonal spheres there correspond two points conjugate with respect to $C$.

15) Generalize the definitions and results of Exercises 12 to 14 to the case where $A$ is a Pythagorean field and where there exists an orthonormal basis for $\Phi$.

16) Let $A$ be a commutative field, $V$ a vector space over $A$ of odd dimension $2r + 1$, $F$ the product space $A \times V$, $\Psi'$ a nondegenerate alternating form on $F$; in the projective space $\mathbf{P}(F)$, of dimension $2r + 1$, one says that the set $C_0$ of lines which are the canonical images of the totally isotropic planes of $F$ (for $\Psi'$) is the linear (projective) complex associated with $\Psi'$.

Suppose in what follows that $A$ is maximal ordered; let $\Phi$ be a positive non-degenerate symmetric form on $V$. Let $D$ be the right orthogonal to $V$ (for $\Psi'$) in $F$, which is contained in $V$, and let $H$ be the hyperplane orthogonal to $D$ (for $\Phi$) in $V$; in $F$, $H$ is a non-isotropic subspace for $\Psi'$, whose orthogonal for $\Psi'$ is therefore a plane supplementary to $H$ and containing $D$. In the affine space $E = \{1\} \times V \subset F$, the set $C$ of the intersections with $E$ of the totally isotropic planes of $F$ (for $\Psi'$) not contained in $V$ is again called the (affine) linear complex associated with $\Psi'$; the right $\Delta = P \cap E$ is called the axis of the linear complex $C$ (for the Euclidean space structure defined on $E$ by the metric form $\Phi$).

a) Now prove that, in $E$, every translation equal to a direction vector of $\Delta$ (Chapter II, 2nd ed., App. II, No. 3) leaves $C$ invariant (cf. § 4, Exercise 6).

b) Let $(e_i)_{0 \leq i \leq 2r}$ be an orthonormal basis of $V$ for $\Phi$, such that $e_0 \in D$ and that $\Psi'(e_{2i-1}, e_{2i}) = \rho_i > 0$ for $1 \leq i \leq r$, $\Psi'(e_j, e_k) = 0$ for pairs of indices which are not of the form $(e_{2i-1}, e_{2i})$ or $(e_{2i}, e_{2i-1})$ (§ 7, No. 3, Prop. 6). Take in the affine space E an origin $a \in \Delta$, and put $\Psi(a, e_0) = \rho_0$. Let $x$ be a vector belonging to the plane generated by $e_{2i-1}$ and $e_{2i}$, and let $y$ be the vector of this same plane such that $\Phi(x, y) = 0$, $\Phi(y, y) = 1$ and that $x \wedge y = \lambda e_{2i-1} \wedge e_{2i}$ with $\lambda > 0$. Let $E_i$ be the 3-dimensional affine linear variety generated by the points $a, a + e_0, a + e_{2i-1}, a + e_{2i}$ in E, and let $R_x$ be the intersection of $E_i$ and the affine hyperplane (in E) generated by the lines of C containing the point $a + x$. Show that the direction of the lines orthogonal (for $\Phi$) to the plane $R_x$ in $E_i$ is a line $L_x$ in the plane $Ae_0 + Ay$, such that if $\theta = (\widehat{D}, \widehat{L_x})$, one has

when the plane $Ae_0 + Ay$ is oriented in such a way that $e_0 \wedge y$ is positive.

¶ 17) a) Let A be a commutative field, $J : \xi \to \overline{\xi}$ an involutive automorphism of A, E a 2-dimensional vector space over A, $\Phi$ a non-degenerate (non-alternating) hermitian sesquilinear form on E, $(e_1, e_2)$ an orthogonal basis of E for $\Phi$ (§ 6, No. 1, Th. 1), such that

$(\alpha$ and $\beta$ belonging to the field K of invariants of J); put $\gamma = \beta / \alpha$. Identify the point $\xi_1 e_1 + \xi_2 e_2 \in E$ with the element $\xi_1 + \xi_2 \rho$ of the ring B defined by the conditions $\rho^2 = -\gamma$, $\rho \xi = \overline{\xi} \rho$ for $\xi \in A$ (§ 3, Exercise 4 a)). For every $x = \xi_1 + \xi_2 \rho \in B$, put $\tilde{x} = \overline{\xi_1} - \xi_2 \rho$ and $N(x) = x \tilde{x} = \tilde{x} x$, so that $x \to \tilde{x}$ is an involutive antiautomorphism of the algebra B (over K) and that one has $\Phi(x, x) = \alpha N(x)$. Show that every similarity for $\Phi$ whose determinant is equal to the multiplier (also called a direct similarity) can be written in one and only one way $x \to xy$, where $y$ is a non-isotropic vector of E, and that its multiplier is $N(y)$.

b) Suppose first that J is the identity, hence $K = A$. If A is of characteristic $\neq 2$, thus recover the results of No. 1. Develop the corresponding theory when A is of characteristic 2 (cf. § 4, Exercise 14; one will distinguish two cases, according as $\gamma$ is or is not a square in A).

c) Suppose $J \neq 1$, so that A is a separable quadratic extension of K. Then $\Phi$ necessarily satisfies condition (T) (§ 4, No. 2 and Exercise 1); if $\Phi$ is of index 0, B is a reflexive field with center K (Chapter VIII, § 11, Exercise 4), and if $\Phi$ is of index 1, B is isomorphic to $\mathbf{M}_2(K)$.

d) Suppose $J \neq 1$ and A of characteristic $\neq 2$; then $A = K(\theta)$ with $\theta^2 = -\delta \in K$. If S is the group of direct similitudes for $\Phi$, H the group of homotheties in E, with ratio $\neq 0$ and belonging to K (a group isomorphic to $K^*$), now prove that the group S/H is isomorphic to the group of rotations $O^+(Q)$, where Q is a nondegenerate quadratic form on a vector space F of dimension 3 over K, such that there exists an orthogonal basis $(f_1, f_2, f_3)$ of F for which one has

(cf. § 9, Exercise 15).

¶ 18) a) Let $A$ be a commutative field, $\xi \to \bar{\xi}$ an involutory automorphism of $A$, $E$ a vector space of even dimension $2m$ over $A$, $\Phi$ a nondegenerate Hermitian form of index 0 on $E$, satisfying condition (T), $\Delta$ the discriminant of $\Phi$ with respect to a basis of $E$. Let $M(\Phi)$ be the group of multipliers of similitudes for $\Phi$ ($§ 4$, Exercise 8). Now prove that $M(\Phi)$ is a subgroup of the multiplicative group of elements of $A$ of the form $\alpha \bar{\alpha} - (-1)^m \beta \bar{\beta} \Delta$. (Argue by induction on $m$, using Exercise 17, together with the following two remarks: 1° if $u$ is a similitude with multiplier $\mu$, $x$ a vector of $E$, $y = u(x)$ and $z = u(y)$, there exists a unitary transform $v$ such that $v(y) = y$ and $v(z) = \mu x$; 2° if $\alpha, \beta, \lambda$ are three elements $\neq 0$ of $A$ such that there exist $a, b, c, d$ in $A$ satisfying the conditions $\lambda = a \bar{a} + \alpha c \bar{c}$, $\lambda = b \bar{b} + \beta d \bar{d}$, then there exist $s, t$ in $A$ satisfying the condition $\lambda = s \bar{s} - \alpha \beta t \bar{t}$.

b) Let $K$ be a maximal ordered field, $K_1 = K((t_1))$ the field of formal power series in an indeterminate $t_1$, with coefficients in $K$ (Chapter IV, $§ 5$, No. 7), $A = K_1((t_2))$ the field of formal power series in a second indeterminate $t_2$, with coefficients in $K_1$. Let $E$ be a vector space of dimension 6 over $A$, $Q$ a nondegenerate quadratic form on $E$, such that there exists an orthogonal basis $(e_i)$ for which one has

Show that there exists no similarity for $Q$ with multiplier $t_1 t_2$.