Exercise 32, § 2

Let A be an algebraically closed field of characteristic 0. a) The only elements of finite order in the group Aut(A) of automorphisms of A are the identity element and elements of order 2 (involutions of A) ; these latter correspond bijectively to the maximal ordered subfields E of A such that $[A : E] = 2$, that is $A = E(i)$ (cf. Ex. 31). If a is an involution of A and E is the fixed subfield of a , then the group Aut(E) of automorphisms of E is isomorphic to the quotient $Z(\sigma)/\{e, \sigma\}$, where $Z(\sigma)$ is the centraliser of $\sigma$ in Aut(A). b) If A is algebraic over its prime field Q then any two involutions of A are conjugate in Aut(A) and the centraliser of an involution a is $\{e, a\}$ ; the group Aut(A) and the set of involutions of A have the cardinality of the continuum, and the centre of Aut(A) is $\{e\}$. c) If A has finite transcendence degree over Q, show that the centraliser $Z(\sigma)$ of any involution a of A is countable (observe that the fixed subfield E of a is countable, and any automorphism of E which fixes the elements of some transcendence basis for E over Q is the identity). If deg . tr_QA = n, show that there exist at least n + 1 conjugacy classes of involutions of A (if E $\subset A$ is maximal orderable with $[A : E] = 2$, consider the transcendence degree over Q of the subfield of E consisting of elements fixed by Aut(E), and use Ex. 24). d) If A has infinite transcendence degree m over Q, show that there exist involutions $\sigma$ of A such that Card$(Z(\sigma)) =$ Card$(\mathrm{Aut}(A)) = 2^m$ (use Ex. 24 to show that there exist maximal orderable fields E such that deg . tr_QE = m, such that for any subset M of any transcendence basis B for E over Q, there exists an automorphism $u_M$ of E such that $u_M(x) \neq x$ for $x \in M$ and $u_M(x) = x$ for $x \in B - M$). e) If deg . tr_QA = 1, show that there is only one conjugacy class of involutions of A whose centralisers are infinite. (If E is a maximal orderable extension of Q of transcendence degree 1 such that Aut(E) is nontrivial, use Ex. 24 to show that E cannot be comparable to Q ; if t $\in E$ is infinitely large with respect to Q, show that so is $u(t)$ for any automorphism $u$ of E, and that conversely if $t'$ is infinitely large with respect to Q then there exists an automorphism $u$ of E such that $u(t) = t'$, using Ex. 15 of VI, p. 40.) f) Let $u$ be an automorphism of A which commutes with all the involutions of A. Show that if a $\in A$ is transcendental over Q then a and $u(u)$ are algebraically dependent over Q. (Consider two elements b, c of A such that $b^2 = a, c^2 = -u(a)$ and observe that there exists a maximal orderable extension E $\subset A$ such that $[A : E] = 2$ and b, c are in E ; obtain a contradiction by noticing that $u(E) = E.$) g) Deduce from f) that $u(a) = a$ for every element a $\in A$ transcendental over Q (consider a maximal ordered extension E of $\mathbf{Q}(a)$ contained in A, algebraic over $\mathbf{Q}(a)$ and comparable to Q, using Ex. 24). Conclude that u is necessarily the identity (observe that if a is algebraic over $\mathbf{Q}$ and $b$ is transcendental over Q then $ab$ is transcendental over Q).