Corollary 3

Let Q be a non-degenerate quadratic form on a finite-dimensional vector space E over an algebraically closed field A. The group of metric automorphisms of E permutes transitively the non-isotropic subspaces of given dimension of E.

This follows immediately from Theorem 1 and Corollary 2 of Proposition 3.

Exercises. — 1) a) Let K be a field of characteristic 2, J : ξ → ¯ξ an involutory antiautomorphism of K, Z the center of K. Show that if the restriction of J to Z is not the identity, every element μ of K such that ¯μ = μ is of the form λ + ¯λ (observe that there is an element ρ ≠ 0 in Z which can be written in the form ζ + ¯ζ, with ζ ∈ Z); every hermitian form on a vector space over K then satisfies condition (T).

b) Give examples of fields of characteristic 2, admitting an involutory antiautomorphism ξ → ¯ξ distinct from the identity map, and for which there are elements $\mu = \bar{\mu}$ which are not of the form $\lambda + \bar{\lambda}$ (cf. Chapter VIII, § 11, Exercise 4).

2) Let A be a field, E a vector space over A, $\Phi$ (resp. Q) a non-degenerate $\varepsilon$-hermitian form on E, satisfying condition (T) (resp. a non-degenerate quadratic form on E, Φ then denoting the symmetric bilinear form associated with Q).

a) Show that in order that a plane P \subset E be isotropic (resp. singular) and not totally isotropic (resp. not totally singular), it is necessary and sufficient that it contain only one isotropic (resp. singular) right (cf. Exercise 14 e)).

b) Suppose that dim $E \geqslant 3$, and that there exist in E isotropic vectors $\neq 0$. Show that if P is a non-totally isotropic plane in E, there exists a non-isotropic vector subspace V \subset E, of dimension 3, containing isotropic vectors $\neq 0$, and such that $P \subset V$.

3) The hypotheses being the same as in Exercise 2, show that if dim $E \geqslant 3$, every isotropic right in E is the intersection of two non-isotropic planes.

4) The hypotheses are those of Exercise 2, and suppose in addition that E is finite-dimensional.

a) If the index $\nu$ of $\Phi$ (resp. Q) is $\geqslant 1$, show that for every isotropic (resp. singular) vector $a \neq 0$ in E, there exists a basis $(e_i)$ of E consisting of isotropic (resp. singular) vectors, such that $e_1 = a$ (cf. Exercise 14 e)).

b) Let V, W be two totally isotropic (resp. totally singular) subspaces of the same dimension $r \leqslant \nu$; show that there exist two maximal totally isotropic (resp. totally singular) subspaces V_1, W_1, such that V \subset V_1, W \subset W_1, and $V_1 \cap W_1 = V \cap W$. (If $U = V \cap W$, reason in $U^0/U$).

c) Let V, W, $V_1, W_1$ be four totally isotropic (resp. totally singular) subspaces of the same dimension, such that $V + W$ and $V_1 + W_1$ are non-isotropic. Show that there exists a metric automorphism $u$ of E such that $u(V) = V_1$ and $u(W) = W_1$.

d) Let $f$ be a linear form on E, $\alpha$ an element of A of the form $\lambda + \varepsilon \bar{\lambda}$ (resp. an element of $\Lambda$). Consider the sesquilinear form on E

(resp. the quadratic form

Show that if $\Phi_1$ (resp. $Q_1$) is non-degenerate and if $\nu_1$ denotes its index, one has $|\nu_1 - \nu| \leqslant 1$.

5) a) Let B be a ring, $\xi \to \bar{\xi}$ an involutory antiautomorphism of B, $\varepsilon$ an element of the center of B such that $\varepsilon \bar{\varepsilon} = 1$. Show that if $\beta$ is an invertible element of B such that $\beta + \varepsilon \bar{\beta} \neq 0$, there exists an invertible element $\mu \neq 1$ in B, such that $\mu(\beta + \varepsilon \bar{\beta})\mu = \beta + \varepsilon \bar{\beta}$. (Show that one can take $\mu$ such that $\mu \beta \bar{\mu} = \beta$).

b) Let A be a field, E a vector space over A, $\Phi$ a non-degenerate $\varepsilon$-hermitian sesquilinear form on E, satisfying (T). Show that if $\Phi$ is not alternating, for every non-isotropic hyperplane $H$ of E, there exists a metric automorphism of E, distinct from the identity, leaving every element of $H$ invariant (use a).

6) The hypotheses being those of Exercise 2, let $a$ be an isotropic vector $\neq 0$ in $E$ (resp. a nonsingular isotropic vector (note that such vectors exist only if $A$ is of characteristic 2)). Let $\lambda \in A$ such that $\lambda + \varepsilon \bar{\lambda} = 0$ (resp. $\lambda = (Q(a))^{-1}$); show that the transvection $x \to x + \Phi(x, a)\lambda a$ (Chapter II, § 6, Exercise 7) is a metric automorphism of $E$; converse.

7) The hypotheses being those of Exercise 2, let $G$ be the group of metric automorphisms of $E$. Show that the only semilinear bijections of $E$ onto itself which commute with all the elements of $G$ are the homotheties of $E$, except in the following three cases: dim $E = 2$, $G$ is the group of metric automorphisms corresponding to a quadratic form of index 1 on $E$, and $A$ is one of the three fields $\mathbf{F}_2, \mathbf{F}_3$ or $\mathbf{F}_4$. (Use Exercises 5, 6 and 3; examine separately the case of a quadratic form on a vector space of dimension 2).

*8) Let $A$ be a field, $E$ a finite-dimensional vector space $> 0$ over $A$, $\Phi$ a non-degenerate $\varepsilon$-hermitian sesquilinear form on $E$, satisfying (T). Let $M(\Phi)$ be the group of multipliers of the similarities of $E$ for $\Phi$ ($§ 6, \mathrm{n}^o\ 5$).

a) Let $V_1, V_2$ be two vector subspaces of $E$, of the same dimension, and let $\Phi_1, \Phi_2$ be the restrictions of $\Phi$ to $V_1, V_2$ respectively. For there to exist a similarity $u$ such that $u(V_1) = V_2$, it is necessary and sufficient that there exist $\alpha \in M(\Phi)$ such that $\Phi_2$ is equivalent to $\alpha \Phi_1$ (use Witt's theorem).

b) Let $(F, F', G)$ be a Witt decomposition of $E$ ($\mathrm{n}^o\ 2$), and let $\Phi_0$ be the restriction of $\Phi$ to the non-isotropic subspace $G$. Show that $M(\Phi) = M(\Phi_0)$ if $G \neq \{0\}$. (Use Witt's theorem and Proposition 2 of $\mathrm{n}^o\ 2$).

c) Now prove that if the index $v$ of $\Phi$ is such that dim $E = 2v$, $M(\Phi)$ is the group of elements $\zeta \neq 0$ of the center of $A$ such that $\bar{\zeta} = \zeta$. If dim $E = 2v + 1$, $M(\Phi)$ is the group of elements of the form $\rho \bar{\rho}$, where $\rho$ runs through the multiplicative group of the elements $\neq 0$ of the center of $A$ (use Witt's theorem). (Cf. § 10, exerc. 18).*

9) Let $A$ be a commutative field, $E$ a vector space over $A$, $Q$ a non-degenerate quadratic form on $E$. A similarity for $Q$ is any automorphism $u$ of $E$ such that there exists an element $\alpha \neq 0$ of $A$ for which $Q(u(x)) = \alpha Q(x)$ for all $x \in E$; $u$ is then also a similarity for the bilinear form associated with $Q$. Assuming the dimension of $E$ to be finite and $> 0$, state and exercise prove, for the similarities relative to $Q$, the analogues of the results of exerc. 8.

*10) Let $A$ be a field, $E$ a vector space over $A$ of dimension $> 2$, $\Phi_1$ (resp. $\Phi_2$) a non-degenerate sesquilinear form $\varepsilon_1$-hermitian (resp. $\varepsilon_2$-hermitian) on $E$ for an involutory antiautomorphism $J_1$ (resp. $J_2$) of $A$, satisfying condition (T). Now prove that if the group of metric automorphisms of $E$ for $\Phi_1$ is a subgroup of the group of similarities for $\Phi_2$, there exists $\alpha \in A$ such that $\Phi_2 = \Phi_1 \alpha$ (use exerc. 5 b) and 6).

Exercise prove the analogous property when A is assumed commutative, and when $\Phi_1$ and $\Phi_2$ are replaced in the statement by two non-degenerate quadratic forms $Q_1, Q_2$ on E.*

11) Let A be a ring, J an involutory antiautomorphism of A, E an A-module admitting a finite basis $(e_i)$, $\Phi$ a non-degenerate $\varepsilon$-hermitian form on E, R the matrix of $\Phi$ with respect to $(e_i)$; the group of metric automorphisms of $\Phi$ is identified with the group G of invertible matrices U such that $^tU . R . U^j = R$.

a) Suppose that there exists a matrix P such that $R = ^tP + \varepsilon P^j$. Now prove that for every matrix S such that $^tS + \varepsilon S^j = 0$, and such that $P + S$ is invertible, $U = (^tP^j - \varepsilon^j S)^{-1}(P + S)$ belongs to G, and $\varepsilon I + U$ is invertible. Conversely (show that for every matrix $U \in G$ such that $\varepsilon I + U$ is invertible, we have

b) Show that the condition in a) is satisfied when $\Phi$ satisfies condition (T). Case where in A the equation $2\xi = \alpha$ has one and only one solution for every $\alpha \in A$.

¶ 12) Let A be a commutative field, E a finite-dimensional vector space of dimension n over A, $\Phi$ a nondegenerate $\varepsilon$-Hermitian sesquilinear form on E.

a) Let u be an endomorphism of E; show that if the $r_i (1 \leq i \leq m)$ are the similarity invariants of u (Chapter VII, § 5, No. 1, Def. 1), the similarity invariants of the adjoint $u^*$ of u with respect to $\Phi$ are the polynomials $\bar{r}_i (1 \leq i \leq m)$, where $\bar{r}_i$ is obtained from $r_i$ by applying the automorphism J to each coefficient (cf. Chapter VII, § 5, exerc. 2). For every monic irreducible polynomial $p \in A[X]$ dividing the minimal polynomial of u, let $F_k(u, p)$ be the kernel of $(p(u))^k$ in E, and let $F(u, p)$ be the union of the $F_k(u, p)$ for all integers $k > 0$. Show that if p and q are distinct monic irreducible polynomials dividing the minimal polynomial of u, the subspaces $F(u, p)$ and $F(u^*, \bar{q})$ are orthogonal (use Bezout's identity). Finally, if G is a vector subspace of E such that $u(G) \subset G$, we have $u^*(G^0) \subset G^0$.

b) Suppose that $uu^* = u^*u$ (the case where u is said to be a normal endomorphism for $\Phi$; cf. § 7, No. 3); show that we then have $u^*(F_k(u, p)) \subset F_k(u, p)$ for every k, and consequently $u^*(F(u, p)) \subset F(u, p)$. If we put $G(p, \bar{q}) = F(u, p) \cap F(u^*, \bar{q})$, show that E is the direct sum of the subspaces $G(p, \bar{q})$, and that $G(p, \bar{q})$ and $G(p_1, \bar{q}_1)$ are orthogonal if $p \neq q_1$ or if $p_1 \neq q$; in particular $G(p, \bar{q})$ is totally isotropic if $p \neq q$. Show that $G(p, \bar{p})$ is reduced to 0 or is nonisotropic, and that if $p \neq q$, no nonzero vector of $G(p, \bar{q})$ is orthogonal to $G(q, \bar{p})$ (use the fact that $\Phi$ is nondegenerate); deduce that if $p \neq q$, $G(p, \bar{q})$ and $G(q, \bar{p})$ are totally isotropic subspaces of the same dimension, and that $G(p, \bar{q}) + G(q, \bar{p})$ is nonisotropic.

c) Suppose that $J$ is not the identity or that $A$ is not of characteristic 2, and that $u^* = u$. Let $\mathscr{M}$ be the set of non-isotropic subspaces $M \subset G(p, \bar{p})$, stable under $u$ (hence submodules of the $A[X]$-module $E_u$ (Chapter VII, § 5, No. 1)). Now prove that if $M$ is a minimal element of $\mathfrak{M}$, $M$ is an indecomposable submodule of $E_u$ (Chapter VII, § 4, No. 7). (Suppose that $M$ is the direct sum of an indecomposable submodule $M_1$ and a submodule $M_2 \neq \{0\}$, the minimal polynomials $p^h$ and $p^k$ of the restrictions of $u$ to $M_1$ and $M_2$ respectively being such that $h \geq k$. Observe then that $M_1$ is necessarily isotropic and that every $z \neq 0$ in $M_1$ such that $p(u).z = 0$ is orthogonal to $M_1$ (use the fact that every submodule of $M_1$ is monogenic); write $z = (p(u))^{h-1}.x$ and that $z$ is not orthogonal to $M_2$, and deduce that necessarily $k = h$. Now prove that there exists an indecomposable submodule $N_2$ of $M_2$ such that $p^h$ is the minimal polynomial of the restriction of $u$ to $N_2$, and that $M_1 + N_2$ is non-isotropic; conclude that $M_2 = N_2$. Finally, if $y \in M_2$ is not orthogonal to $z$, consider the submodule $P$ of $M$ generated by $w = x + \lambda y$, where $\lambda \in \mathbf{A}$, and prove that $\lambda$ can be chosen so that $P$ is non-isotropic, by proving that $\Phi((p(u))^{h-1}.w, w) \neq 0$; which leads to a contradiction).

d) Deduce from c) that $G(p, \overline{p})$ is the direct sum of pairwise orthogonal indecomposable submodules $H_i$. If $p^h$ is the minimal polynomial of the restriction of $u$ to $H_i$, and if $d$ is the degree of $p$, prove that there exists in $H_i$ a totally isotropic subspace of dimension $d.[h/2]$. Case where $E$ contains no isotropic vector $\neq 0$ (cf. § 7, No. 3).

e) State and prove the analogous properties to those of c) and d) when $u^* = -u$ or $u^*u = 1$.

f) Give an example where $n = 4$, $\Phi$ is symmetric and of index 2, $p = \overline{p} = X - 1$, $u$ is normal, $E = G(p, \overline{p})$, but $E$ is not the direct sum of minimal submodules of $\mathfrak{M}$, and where there exists an eigenvector of $u$ which is not an eigenvector of $u^*$ (cf. § 7, No. 3).

13) The hypotheses are those of Exercise 2, and suppose in addition that $E$ admits a countable basis $(e_n)$. Let $F$ be a totally isotropic (resp. totally singular) subspace of $E$ such that $F^{00} = F$; now prove that there exists a totally isotropic (resp. totally singular) subspace $F'$ such that: $1^\circ \ F \cap F' = \{0\}$; $2^\circ$ there exists a basis $(a_m)_{m \in I}$ of $F$ and a basis $(a'_m)_{m \in I}$ of $F'$ (the interval of $\mathbf{N}$ with origin 0) such that $\Phi(a_i, a'_j) = \delta_{ij}$ for every pair of indices; $3^\circ$ $(F + F')^{00} = F + F'$ and $E$ is the direct sum of $F + F'$ and $G = (F + F')^0$. (Form by induction an increasing sequence $(L_n)$ of non-isotropic subspaces, whose union is $E$, such that $\dim L_{n+1} - \dim L_n = 2$, and apply Proposition 2 of No. 2 to each of the $L_n$; to form this sequence, consider, for each $n$, the smallest integer $k$ such that $e_k \notin L_n$, and use Exercise 9 b) of § 1).

14) Let $A$ be a field of characteristic 2, $E$ a finite-dimensional vector space of dimension $n$ over $A$, $\Phi$ a non-degenerate hermitian form on $E$, not necessarily satisfying condition (T).

a) Show that the set $V$ of the $x \in E$ such that $\Phi(x, x)$ is of the form $\alpha + \overline{\alpha}$ is a vector subspace of $E$.

b) Let $V_1 = V \cap V^0$, $q = \dim V_1$, $V_2$ a supplementary space to $V_1$ with respect to $V$, $V_3$ a supplementary space to $V_1$ with respect to $V^0$. Now prove that there exists a base $(e_i)_{1 \leq i \leq 2q}$ of $(V_2 + V_3)^0 = V_2^0 \cap V_3^0$ such that the vectors $e_1, \ldots, e_q$ form a base of $V_1$ and that $\Phi(e_i, e_{q+j}) = \delta_{ij}$ for $1 \leq i \leq q, 1 \leq j \leq q$.

c) Let $G(\Phi)$ be the group of metric automorphisms of $E$ (for $\Phi$). Now prove that for every $u \in G(\Phi)$, we have $u(x) = x$ for every $x \in V^0$.

d) For every $u \in G(\Phi)$, we have $u(V) = V$; let $u_v$ be the restriction of $u$ to $V$, and let $G_v$ be the group formed by the $u_v$. Now prove that: $1^o$ the kernel of the homomorphism $u \to u_v$ from $G(\Phi)$ onto $G_v$ is commutative; $2^o$ if $\Phi_2$ is the restriction of $\Phi$ to $V_2$ and $G(\Phi_2)$ the group of metric automorphisms of $V_2$ for $\Phi_2$, there exists a homomorphism from $G_v$ onto $G(\Phi_2)$ whose kernel is commutative (use $b$ and $c$).

e) Suppose that $A$ is commutative and $J$ is the identity; let $E$ be a vector space of dimension 3 over $A$, $(e_i)_{1 \leq i \leq 3}$ a base of $E$, $\Phi$ the non-degenerate symmetric form on $E$ whose matrix with respect to $(e_i)$ is

Show that all isotropic vectors in $E$ are contained in the hyperplane generated by $e_2$ and $e_3$ (cf. exerc. 4 a)). Give an example of a non-isotropic plane containing only one isotropic right (cf. exerc. 2 a)). Show that there is no automorphism $u \in G(\Phi)$ such that $u(e_1) = e_1 + e_2$, though we have $\Phi(e_1, e_1) = \Phi(e_1 + e_2, e_1 + e_2)$.