1947] AUTOMORPHISMS OF THE FIELD OF COMPLEX NUMBERS

AUTOMORPHISMS OF THE FIELD OF COMPLEX NUMBERS

By H. KESTELMAN

[Received 13 October 1947.—Read 16 October 1947.—Received in revised forni 11 June 1948]

1. Introduction. A function <j>{z) defined for all complex numbers z sothat for every z1 and z2

(1) 0(Z!.+22) = <£(Zi)+<£(22)and

(2) <f>(Zlz2) = <j>{zx)<j>{z2),and so that

(3) the equation <£(z) = £ has a solution for every complex number £

is said to define an automorphism of the field of complex numbers. Obviousfunctions satisfying these conditions are <j>{z) = z, <f>(z) — z. (It is plainthat if <f>(u) = <f>(v), then <f>(u—v) = 0, and consequently, by (2), any <f>satisfying (1) and (2) which is not identically zero cannot assume anyvalue twice.) The problem of the existence of non-trivial solutions waspropounded by C. Segre in 1889 (1, p. 288) in connexion with the problemof existence of non-projective collineations in a bicomplex plane. Coolidge(2, p. 38) referred to it as an unsolved problem in 1921, though in 1945he mentioned (3, p. 63) reported solutions of which he had not seen proofs.The corresponding problem in the field of rear numbers had been solvedby Darboux (4) in 1880; he proved that <j>(x) = x is the only solution,thereby completing the proof of von Staudt's theorem that a real collinea-tion is necessarily projective.

The existence of non-trivial solutions of the functional equation (1) wasdemonstrated by Hamel (5) in 1905, but no progress seems to have beenmade with Segre's problem until Lebesgue's paper (6), in the form of aletter to Segre, appeared in 1907. This brilliant contribution to the prob-lem ignores condition (3) above and is in places obscure, but it containsthe essentials of a complete solution; in a subsequent article (7) Lebesguerefers to the 1907 note as a "reponse partielle" to Segre's problem.

The original version of this paper had for its main aim the completionSER. 2. VOL. 53. B

2 H. KESTELMAN [16 Oct.

of Lebesgue's argument by such methods as would occur to one inexpertin algebra. It has since been pointed out by the referees that the existenceof non-trivial automorphisms of the complex numbers is deducible fromgeneral theorems due to Steinitz. I am, above all, indebted to Dr. R. Radowho has summarized the argument in a form that will be at once compre-hensible and satisfying to an algebraist. I give Dr. Rado's account, inhis own words, in § 3. The theorems quoted from Steinitz and van derWaerden are, however, not needed in their full generality as far as ourproblem is concerned. Accordingly, in § 4, the argument is formulated ina more concrete way; this is less revealing than the general approach, butit is quite elementary and, except for an appeal to a few basic results inthe theory of extension of number fields, quite self-contained.

The arguments used by Lebesgue and by Steinitz rely on Zermelo'stheorem that the continuum can be well ordered, and the non-trivialautomorphisms which are subsequently "constructed" are defined bymeans of transfinite induction. Both Hamel and Lebesgue had provedthat every non-trivial solution of (1) alone must be non-measurable, andOstrowski's researches on the same equation (8) have since proved thatevery solution of (1) in the field of real numbers which is not trivial mustfluctuate wildly in every linear set of positive measure. In the secondpart of this paper we prove, among other results, that every functionwhich defines a non-trivial automorphism of the complex numbers trans-forms every bounded set (in the Argand plane) into a set of Lebesguemeasure zero or else into a non-measurable set. It is therefore not sur-prising that a complete solution of Segre's problem should be virtual.

From a geometric point of view, the function <f>(z) defines an interestingtransformation of the bicomplex plane (the set of all points (x, y) wherex and y are complex numbers). If the point {<f>(x), <f>(y)} is assigned to thepoint (x,y), the transformation is a collineation (in the simple sense thatit is 1-1 and that collinear points are transformed into collinear points,both ways) since the algebraic condition for three points to be in line will,by (1) and (2), ensure the collinearity of the transforms. Further, if fourordered collinear points have a cross-ratio p, their transforms have cross-ratio <f>(p) by (1) and (2). The collineation is neither projective nor anti-projective, though it will of course, as a collineation, preserve harmonicrelations (in other words, <f>{p) = p whenever p is rational). The trans-formation can be extended to cover the points of a line at infinity byassigning the point { (a ),<f>(x2),<f>(x3)} to the point (x1}x2ix3); the line atinfinity, x3 = 0, is then a fixed one. In spite of the fact that the trans-formation is discontinuous at every point of the plane, it has a number ofunexpectedly normal properties. It transforms every conic into a conic,

1947] AUTOMORPHISMS OF THE FIELD OF COMPLEX NUMBERS .*$

every circle into a circle; this follows from (1) and (2) since the alge-braic curve with equation ^arsx

rys = 0 is transformed into the curve2 (f>((irs)x

ry8 = 0. In particular, if the ars are all integers, the curve willbe fixed since <f>(n) — n for every integer n. The circular lines x2-\-y2 = 0.for example, are fixed or permuted. Parallelism and perpendicularity oflines are both invariant, and so in fact is any geometric property of analgebraic curve which can be expressed by the vanishing of a polynomialin the arfS with integer coefficients: for instance, a parabola (h2 — ab, A ^ 0)is transformed into a parabola.

The question of what restriction need be imposed on <f>, in addition to(1), (2), and (3), to exclude the non-trivial solutions is considered in § 7.Coolidge (2) assumes continuity, but much weaker restrictions suffice, inview of the extreme fluctuation, already referred to, of any non-trivial <f>.

2. Notation. Z is the set of all complex numbers; it also sometimesdenotes the Argand plane (a real plane in which the complex numberx-\-iy is represented by the point with real coordinates (x,y)). In § 7 wedeal with a bicomplex plane 3 which consists of all (x, y), x and y beingany complex numbers. The set of all z in Z which satisfy \z—y\ < 8 iscalled the circle in Z with centre y and radius 8.

R is the set of all rational and ^? the set of all real numbers. If f(z) is afunction defined in a set E, f(E) is the set of all numbers f(z) for whichz e E; /-1(AS) is the set of all z for which f(z) e S, S being-any set of complexnumbers.

mi E is the interior Lebesgue measure of E (upper bound of Lebesguemeasures of closed subsets of E), \E\v& the exterior Lebesgue measure ofE (whether the measure is linear or plane will be made clear by the con-text). If | JE7 I = 0, we say E is null.

The algebraic terminology follows (9). In particular, if E c Z, B[E]consists of all numbers which can be expressed as finite sums of productsof a finite number of terms in E-\-R, and R{E) is the field of quotientsof such numbers. A number z is algebraic in a field K if z is a zero ofa not identically vanishing polynomial over K. A function <f>(z) (notidentically zero) which satisfies (1) and (2) for all zy and z2 in a field Kwill be called a Segre function, or an isomorphism, on A", a trivial one if<f)(z) = z or <f>(z) = z. If tp(z) is an isomorphism on H D K and 0(z) = <f>(z)in K, we say </r extends <f>. If K is algebraically closed, i.e. every poly-nomial over K is a product of linear factors, then it is clear that <f>{K) isalso algebraically closed if <f> is a Segre function on K.

The cross-ratio — -/— is denoted by {z^z.?; z3,z±).2 3 Z2l Zi Z2

4 H. KESTELMAN [16 Oct.

3. The existence of non-trivial automorphisms of Z is contained im-plicitly in Steinitz's classical paper (10). The deduction has been sum-marized by Dr. R. Rado as follows:

Z is an extension of R\ hence (10, p. 293, 2) there is a set T c Z suchthat

(4) any finite number of elements of T are algebraically independentover R, and

(5) Z is algebraic over R(T).

T includes more than two elements, for otherwise R{T) and Z would beenumerable. Let x0 E T; then T includes an xY such that x1 ^ x0, xx ^ x0.Let f(x) be a permutation of the numbers of T such that/(a;0) = xv By(4) there is a unique extension of the definition of/ into R(T) such thatf(x) defines an automorphism of R(T). By a general principle in algebra(e.g. 12, vol. 1, p. 42) there exists a field Y 3 R(T) and an extension ofthe definition of/(a;) into Z such that / defines an isomorphism from Zto Y. Since Z is algebraic over R(T) and algebraically closed, it followsthat f(Z) = Y is algebraic over F{R(T)} = R{T) and algebraically closed.Hence (10, p. 287, 9) Y and Z are equivalent extensions of R(T), i.e. thereexists an isomorphism g(y) from Y to Z such that g{t) = t for teR{T).Then h(x) = g{f(x)} defines an automorphism of Z, and

} = y(xi) = xv

Hence h(x) is a non-trivial automorphism of Z.

4. In this paragraph we give a more concrete and self-contained versionof the argument contained in § 3.

4-1. The construction of the set T, which is due to Lebesgue and toSteinitz, is easily described. Let zl, z2,..., z^,... be a well-ordered series,denoted by (S), of all the complex numbers, and let Pa be the set of allzr with r < a. T is defined to consist of all those za which are not algebraicin the corresponding field R(Pa). If there exist complex numbers notalgebraic over R(T), let za be the first such number to occur in (S). Sinceza e T, za is algebraic over R(Pa). Since every z in Pa is algebraic overR(T) so is za (9, p. 389), and this is a contradiction. This proves (4) and(6) of §3.

4-2. L e t / be any permutation of T satisfying f(x0) = xx (§ 3). By (4)of § 3 it follows that if 9 and I/J are rational functions (over R) of indeter-minates xx, x2)..., xn, and they are equal for one set zv z2,..., zn in T, thenthey are also equal for the set/(zj, f(z2),..., f(zn). Hence, if

z — 6(zv z2,..., zn),

1947] AUTOMORPHISMS OF THE FIELD OF COMPLEX NUMBERS 5

where zv z2,..., zn are in T, it follows that the definition

/(*) = *{/(*i)Jfo)>... ,/(*»)}defines a (one-valued) function over R(T). If x = 6(zv z2,..., zn) andy = i{j(zv z2)..., zn) are in R{T), then x-\-y is the value of 0+«A at (zls z2,..., zn)and consequently f{x+y) = f{x) +f(y). Similarly/(xy) =f(x)f(y). More-over, f(z) = xtiz = flf/-1^),..., Z" 1 ^ )} , and so /(tf(T)} = R(T); hence/(z) defines an automorphism, of R{T).

4-3. We now extend/, defined on R{T), to a Segre function <j> on Z\ ifthis is done, <£(Z) will be algebraically closed (like Z) and therefore includeall numbers algebraic in <f>{R{T)}. Since <f>{R(T)} = -R(T'), this means<I>(Z) D !Z and so <f> is an automorphism of Z with ^(a;0) = xv Suppose£1? t,2,..., is a well-ordered series of all numbers in Z—R(T), and suppose<f> has been defined as a Segre function on F^, the extension of R(T) byall £r having r < a. If £a e Fa, <£(£a) will have been defined. Suppose nowthat £a e jPa. Since ^a is algebraic in .Fa (being algebraic in R{T)), thereis a unique irreducible polynomial

2 ^ ^ ^r=0

such that P(Ca) = 0. Let X be chosen to satisfy

r = 0

If p(t) and g(<) are polynomials over jPa with p{£a) = qi^), then > = q(mod P); hence, if

* = i<v£(c r e jF a ) and we define < (a;) = f ^(cr)J^,r=0 r=0

it follows, since <f> is a Segre function on ^a, that <f> is one-valued in jPa(^a),and arguments parallel to those used in 4*2 show that ^ is a Segre function

By the principle of transfinite induction, the above process defines <f> inZ so that <f> is a Segre function on every F^. Since every pair of numbersin Z belong to Fa for some a, <f> is an isomorphism defined on Z whichextends / , and this completes the proof.

4-4. I am indebted to a referee for the suggestion to use Zorn's principle,and also to Dr. J. L. B. Cooper who made some very helpful suggestionsin the same sense. It may be of interest to show how the argument of4-3 is transformed when Zorn's principle (14) replaces the well-orderingtheorem of Zermelo.

P> H. KESTELMAN [16 Oct.

Let / 0 be the automorphism of j?(v'2) denned by /0(V2) = — V2 (anynon-trivial automorphism of a subfield of Z would do instead). We showthat there is an automorphism of Z which extends /0 .

Suppose B is a set of isomorphisms such that any element of B eitherextends or is extended by any other element of B. It is clear that, if Eis the union of the fields of all the elements of B, then (i) E is a field,(ii) if F(z) is defined in E as 6(z), where 6 is any element of B whose fieldof definition includes z, then F is an isomorphism on E, and (iii) F is anautomorphism if every element of B is one.

Now consider the class s/ of all automorphisms Avhich extend /0 . ByZorn's principle, srf has a maximal element /x which cannot be extendedby any element of J / . Our object is attained if H = Z, H^ denoting thefield on which /x is defined. Suppose that G = Z—H^ ^ 0. If £ e G, then£ is algebraic over Hfl, for if this were false we could, following the argu-ment of 4-2, extend /x by an automorphism over H(£). Now consider theclass J of all isomorphisms which extend /u. By Zorn's principle Jf hasa maximal element v. The arguments of 4-2 and 4-3 show that a maximalelement of./ must have Z for its field. Hence v is an isomorphism on Zwhich extends/„. Our object is attained if v(Z) = Z, and since v extends fxit is enough to show that v(Z) D G. This follows at once from the fact thatj/( Z), like Z, is algebraically closed and the fact that G is algebraic in H .

5. It was proved by Hamel (5) that if (f>(x) is real and satisfies (1) forall real zx and z2, then the values assumed by <f>(x) in any interval of 3k,however small, are everywhere dense in (—00,00), except in the trivialoase <f>(x) = xcf>(l). This was extended by Ostrowski (8), who showed thatinterval may be replaced by set having positive interior measure. In thissection we prove a series of similar results for non-trivial Segre functionson Z; they include and extend results of E. Noether (15). These resultshave a bearing on the problem of finding conditions which are sufficientto restrict a Segre function to the trivial cases. From the geometricalpoint of view, there is some interest in considering the character of <f>(z)in so far as it is determined by its values for real z.

LEMMA 1. Suppose that <f>(z") is a Segre function in Z and <f>(z) is con-tinuous relative to & at one point of &, then <j> is trivial.

Proof. By (1), <f>(z) is continuous relative to & at all points of (ft. Since(f>(x) = x whenever x is rational, this means that <f>{x) = x whenever x isreal. By (2), {<f>{i)f = <f>{-l) = - 1 , and so </>{i) = ±i. By (1) and (2)A\ c now conclude that

<f>i.e. <f> is tr ivial .

1947] AUTOMORPHISMS OF THE FIELD OF COMPLEX NUMBERS 7

THEOREM 1. Suppose that <j>(z) is a Segre function in Z. If Ec& andmtE > 0, and <f>(z) is either (i) bounded in E or (ii) real for all z in E,then </> is trivial.

Proof. By a theorem of Steinhaus (11), there is a positive number 8such that every x satisfying — 8 < x < 8 can be written x = a—b, wherea and b belong to E.

If (i) holds, let M be the upper bound of \cf>(z) | in J57; then — hjn < x < hjnimplies \<f>(x)\ < 231/n, and so <f>(z) is continuous relative to 8& at z •-=• 0.Lemma 1 completes the proof.

If (ii) holds, then <j)(x) is real whenever —8 < x < 8, and, since<f)(rx) = r<f>(x) for rational r, <f>(x) must be real throughout &. If now xand h are real, h ^ 0, then

j{x+h*)-4>(x) = <^2) = fo(A)}2 > 0.Thus <j>(x) is monotone in 9t and </>(x) = x whenever x G i2. This implies<f>(x) = a; throughout £%; lemma 1 completes the proof.

THEOREM 2. / / <j>{z) is a non-trivial Segre function in Z, then <(>(&) iseverywhere dense in Z.

Proof. By theorem 1 (ii), there is a real £ with <f>(£) unreal. Hence, ifrx and r2 range through R, the numbers fi<£(£)+?'2, i.e. the numbers<^(rx^-i-r2) are everywhere dense in Z.

THEOREM 3. / / </>(z) is a non-trivial Segre function in Z, and U is anycircle in Z, then the set of real x with <j>(x) e U is non-measurable and haszero interior measure (in &).

Proof. Let H — ^ ~ 1 ( t / ) . Take £ as in the proof of theorem 2 andthen rx and r2, rx ^ 0, so that <t>ixx£-\-r^ e U, i.e. rx£-\-r2 e H. If \H\ = 0it follows, since R is enumerable, that the set of all £ is null and conse-quently that <j>(x) is real almost everywhere in ^ , which contradictstheorem 1 (ii). Hence \H\ > 0. Since <f>{H) is bounded, m^l — 0 bytheorem 1 (i).

THEOREM 4. / / <£(z) is a non-trivial Segre function in Z, and S is anyset of real numbers with mi S > 0, then <f>(S) is everywhere dense in Z.

Proof. There is no loss of generality if we assume S to be closed andbounded and that the lower bound of \x\ for x in S is positive. Let v bethe greatest value of \x\ for x in S.

Suppose first that there is a positive number p such, that \<f>(x) \ > p forall x in 8. Let T be the set of numbers l/x as x ranges through S. PlainlyT is closed, and \<f>(t)\ < 1/p for all t in T; this contradicts theorem 3 ifmT > 0. Now if T is covered by intervals (xr,yr) (r = 1,2,...) having

8 H. KESTELMAN [16 Oct.

\xr\ and \yr\ both greater than ty-1 and xryr > 0, then S is covered byintervals the sum of whose lengths is 2 l^r"1"^1!* which is less than

r

(2v)2 2 (yr—xr)y anc* s o mT = ° implies mS = 0. Thus mT > 0, and it

follows that <f>(8) has points in every circle centred on the origin.Now suppose that <f>(8) has no points in a circle \z—y\ < p. By

theorem 2 there is a real number f with |<£(£)—y\ < \p. Let E be theset of real x for which (#+£) e #. Then raj£ > 0, and if xe E we havel*(aH-£)-yl>p. i-e. l # e ) - H ( | ) - y | > p, as weU as | f l £ ) - y | < */>.Hence |<£(z)| > £p for aM x in E, and this contradicts the case first con-sidered.

THEOREM 5. Suppose that <f> is a non-trivial Segre function in Z and Eis any subset of Z with both interior and exterior points.* Let I be any linein the Argand plane; then l^-^E) is (i) everywhere dense in I, (ii) non-measurable and of zero interior measure. The same is true of ty(E) if <f> isan automorphism.

Proof. First suppose that I is the real axis. If mi{Z^~1(^)} > 0, thenby theorem 4 and the hypothesis on E, < {Z -1(JS/)} includes points of Z—E,which is impossible. Hence m<Z^~1(^) = 0. Replace E by Z—E to getmM-iiZ—E) = 0. This implies (i) and (ii).

If I is the imaginary axis, then <f>{iy) e E means that <j>{i)<f>(y) e E} i.e.(f>(y) ef{E), where f{E) is a linear transformation of E. This reduces tothe first case. Finally, if I has the equation y = ax-\-b (a, b real), then theprojection on the real axis of hj>-x{E) is the set of real x for which<f>{x-\-i(ax-{-b)}, i.e. {<f>(x)<f>(l-\-ia)-\-<f>(ib)} e E\ this again restricts <f>(x) toa linear transformation of E and so reduces to the first case.

If <f>(z) is an automorphism, so is ^~1(z), and we may replace <f> by <£-1

in (i) and (ii).We now consider properties of Segre functions on plane sets in Z.

THEOREM 6. Suppose that <f>(z) is a non-trivial Segre function in Z; then(i) <f>(S) is everywhere dense in Z if mtS > 0; (ii) if E is a set with bothinterior and exterior points, then <j>~x{E) his zero interior measure and itnon-measurable. (The same is true of <f>{E) if <f> is an automorphism.)

Proof. If <f>(S) has no points in some circle U, then Sc<j>-l(Z—U).But (putting E = Z— U in theorem 5) this implies that every line cutsS in a linear set with zero interior measure. This, by Fubini's theorem,implies mt S = 0 and proves (i). If m^-^E) > 0, then by (i) E is every-where dense, and this is impossible since E has exterior points. Hence

= 0. Since E may be replaced by Z—E, we conclude that

* i.e. E and Z—E both possess interior points.

1947] AUTOMORPHISMS OF THE FIELD OF COMPLEX NUMBERS 9

has zero interior measure and is non-measurable. Finally, if <f> isan automorphism we have the desired conclusion as in theorem 5.

Theorem 6 leads to the paradoxical conclusion that, if ikf is a measurablesubset of Z and E is a set which has both interior and exterior points,then \Mj>-x{E) \ = \M<j>(E) \ = mM if<f> is a non-trivial automorphism of Z;this is because ^^(Z—E) and <f>(Z—E) have zero interior measure. Forexample, if {£7n} is a sequence of mutually exclusive circles, the setsM^iUj.) will be a sequence of exclusive subsets of M each having thesame exterior measure as M.

6. In this section we discuss the geometric aspect of the transformationof the Argand plane Z {not the bicomplex plane 3) by a function <j> whichdefines an automorphism of Z.

LEMMA 2. Suppose that S is a subset of Z which includes unreal numbersand which has the properties

(i) if ae S and b e 8, then (a—b) e S,(ii) if ae 8 and r is rational, then ar e 8,

((i) and (ii) hold, for example, if S is a field in Z); then there are threepossibilities, just one of which must be realized:

either 8 = Z, or \8\ = 0, or mt8 = m^Z—8) = 0.Proof. Suppose that mt 8 > 0. By Steinhaus's theorem and (i), 8 con-

tains a circle with its centre at 0. Hence, by (ii) 8 = Z.Now suppose 0 = mi 8 < \8\. We have only to prove that m^Z—8) = 0.

The points of 8 do not lie on a single line through 0; hence, by (i) and (ii),8 is everywhere dense in Z. Suppose now that m^Z—S) > 0, and let £be a point of Z—8 which is the centre of arbitrarily small circles Usatisfying mi{U(Z-S)} > f|C7|, i.e. \US\ < l\U\.

Since £ is a limit point of 8 it follows that there are arbitrarily smallcircles V which have their centres in S and satisfy \VS\ < £|F|. Now itfollows easily from (i) that, for any circle V which has its centre in 8,\VS\ depends only on the radius of V. Since we have just shown thatfor such circles \VS\ < £|F| for arbitrarily small V, it follows that theexterior density of 8 is never unity at any point of S. This contradicts|#| > 0 and so establishes m^Z—S) = 0.

THEOREM 7.* / / <j>(z) is a non-trivial Segre function in Z; then (i) <f>(&)is a proper subfield of Z with cardinal number c, (ii) either <f>(&) is null, orelse m^M) = m^Z—M) = 0.

Proof, (i) That <f>{^) is a field follows from (1) and (2), the rest fromthe fact that the mapping of Z on <f>{Z) is 1-1. (ii) Put S = <j>(&) in

* In connexion with (i), see also M. Souslin (13).

10 H. KESTELMAN [16 Oct.

lemma 2 to get m ^ ( ^ ) = 0. Since <f>{Z-@) c Z-<f>{@), m^Z-M) = 0by the same lemma.

We can now give a picture of the geometric transformation of theArgand plane Z into <f>{Z) when <f> defines an automorphism of Z. Weshow that the <f>(Z) plane is covered by each of two "orthogonal" familiesof sets (corresponding to the lines in the Z-plane parallel to the real andimaginary axes). The set <f>(&) may be used as a basis for the description.

Let Yv and Xg denote the transforms by <f>{z) of the lines y = r\ andx — £ respectively in Z. Plainly Yo = <f>(@) and, since <f>(iz) = ±.i<j>{z), Xo

is the result of rotating Yo about the origin through an angle ±^TT. YV isthe result of translating Fo by the vector 4>(irj) and Xg the result of trans-lating Xo by the vector <f>(£).

Thus (f>(Z) is covered by the system of Xg and also by the system of Yv;through any point iv in <j>(Z) there passes just one Yv set and one Xg set:the former is obtained by translating <}>(&) so that one of its points fallson w, and the latter by rotating the Yv set about w through an angle ±\rr.

If <f>(&) is not null, it follows by theorem 7 (ii) that the transform by<j)(z) of the line y = rj cuts every measurable set E in the <f>(Z) plane ina set whose exterior measure is mE.

It may be noted in passing that, if <f>~l(w) = X(w)-\-iY(w), where X andY are real functions, then these functions are additive (i.e. they satisfyequations such as (1)) and that, since <^~1(ro) = 0t, Y(w) = 0 meansw e Yo. Hence <f>(!%) is the set of all the periods of the function Y(w).

7. Segre's geometrical problem may be stated as follows.

Suppose ihat T(P) is a 1-1 transformation of a bicomplex plane 3 intoitself, which transforms every three collinear points into collinear points: whatis the relation between the cross-ratio of four collinear points and that of theirtransforms by T?

Both Segre and Coolidge conjectured that the two cross-ratios are eitheralways equal or always conjugate complex numbers. If this were so, itwould mean that T is either a projectivity or else the product of a pro-jectivity and the transformation which assigns {zv z2, z3) to the point withhomogeneous coordinates (zvz2,z3). If a collineation T satisfies one ofthese alternatives, we shall write T e S. It was pointed out in § 1 thatevery non-trivial automorphism of Z could be used to define a T such thatTeS. In Coolidge (2) it is usual to impose a condition of continuity on Tin order to make TeS, but we shall see that weaker restrictions suffice.

A collineation T will be said to have the Segre property in a set E ifthe cross-ratio of every four collinear points of E is either always equalto, or is always the conjugate-complex of, that of their transforms under

1947] AUTOMOBPHISMS OF THE FIELD OF COMPLEX NUMBERS 11

T. We observe first that if T has the Segre property in a line I of 3 , thenit has it in the whole of 3 : for, ii Qr (r — 1, 2, 3, 4) are collinear pointsnot on I, project them on to I from some point V, Qr going into Q'r. Theline joining T(V) to T(Qr) cuts T(l) in T(Q'r), and since {Qv Q2; Qz, Q4}and {Q\, Q'2; Q'3, Q'A] are equal, it follows easily that T has the Segre pro-perty in the line Qx Q2 if it has it in I. It is therefore sufficient, as well asobviously necessary, to show that T has the Segre property in lx, the''';u-axis'! of 3> if we wish to conclude T e 8. We suppose, for convenience,that T is extended in the obvious way so as to apply to the points atinfinity of 3- J-^t Pz denote the point (z, 0) and P^, the point at infinity

on lx. Suppose that T{P0) = A, T^) = U, and T{Pm) = / . For everyfinite z we define <f>{z) = {I, A; U, T{PZ)}. [The condition that T shall havethe Segre property on lx is then the condition that <f>{z) = z or <f>(z) = zin Z, i.e. that <f>(z) is a trivial Segre function in Z.] We first show that<f)(z) is a Segre function in Z. Iizx and z2 are in Z, then (PZl+Zt; Po), (PZl; PZ2),(POOJPQO) are mates in an involution on lx. Using Desargues's theorem onthe involution properties of a quadrangle and the fact that incidence oflines is invariant under T, we deduce that the transforms of these elementsare likewise mates in an involution on T(lx). Consequently, since {a, b\ c, t}is, for fixed a, b, c, a linear fractional function of t, it follows that{<f>(z1-\-z2);O}, {<f>{z1);<f>(z2)}} (oo;oo) are mates in an involution of complexnumbers, i.e. that (1) holds. Similarly, since (zx32; 1), (z1;z2), (0;oo) aremates in an involution, we deduce that (2) holds, and so <f)(z) is a Segrefunction in Z.

From what has just been proved, it follows that any set of conditionswliich is sufficient to make a Segre function trivial will at the same timegive a geometrical condition sufficient to make T e 8.' The properties ofnon-trivial Segre functions considered in § 5 make it easy to state suchconditions. For example, if we restrict our attention to the real pointsof lx, it follows from theorem 4 that if lx contains a real interval, or evena real set E with positive interior measure, such that T(E) is not every-where dense in the complex line T(lx), then T e 8. In particular, if T(P)is real or if T(P) is bounded in E, or if T(P) is continuous relative to Eat one point of E, then T e S.

Addendum. Dr. L. Roth has kindly shown me a paper by B. Segreentitled " Gli automorfismi del corpo complesso, ed un problema di CorradoSegre" which has appeared in the Atti della Ace. dei Lincei (Fasc. 5-6,novembre-dicembre 1947, p. 414). The existence of non-trivial auto-morphisms is established there as a deduction from Steinitz's theoremsin much the same way as in § 3 above.

12 H. KESTELMAN [16 Oct.

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University College,London.