A really elementary proof of real Lúroth ’ s

Classical Lifroth theorem states that every subfield 1< of K(t), where t is a transcendental element over K, such that 1< stnictly contains K, must be 1< = K(h(t)), fon sorne non constant element hQ) e K(t). Therefore, E la K-isornorphic to K(t). This result can be proved with elementary algebraic techniques, and therefore it la usual!>’ included in basic courses on fleld theory or algebraic curves. In this papen we study the validity of this resuit under weaker assumptions: namel>’, if E is a subfield of C(t) and 1< strictly contairis R (¡1 the real fleld, C the comnplex ficíd), when doce it hoid that E is isonxorphic to R(t)? Obviously, a necesear>’ condition is that 1< admite an ordening. Here we prove that this condition is also sufflcient, and we cali such statement the Real Lúroth’s Theorenx. There are several ways of pnoving this result (Riemann’s theorem, Hilbert-Hurwitz ¡3]), but we claim tbat proof is really elementary, since it don require just sorne basic background as in the elassical version of Liiroth’s. 1 Real Lfiroth’s Theorem Liiroth’s Titeorem usuall>’ appears in counses on fleld titeor>’ or in courses on algebraic curves, and -~as it is well known— states titat every subfleld *Partiafly supported by CICyTPB92/0498/C0201 (Geometría Real y Algoritmos), Espnit/flra 6846 (Poso), and TIC-1026-CE. ~Partiaflysupported by Univ. Alcalá Proy. 030795. AMS subject classificatioxr 14H05, 141>05. Servicio Publicaciones Univ. Complutense. Madrid, 1997. 284 2’. Recio ¡md J. R. Sendra E of tite fleld K(t) (t transcendental oven K) transcendental oven K (this means, in particular, titat E contains K), is isomorpitic to X(t), see [4]voL II Pp. 515 (i.e E has tite form K(it(t)), fon sorne h(t) E K(t)), or equivalentí>’, that tite fleid of rational functions of ever>’ K-rational plane curve is isomorphic to K(t), see 1 7] vol. 1 pp. 9. Let 110W .1<’ be an algebraic extension of K, and E a subfleld of K’(t) (t transcendental oven It) witich is transcendental oven K. Titen tite natural question of auial>’zing uf E is isomorphic to K (t) anises. Ir particular, if K = 11 auid K’ = ’ stud>’ if E is isomorpitie to 11(t). Cleaní>’, titis is not tnue fon ever>’ E. Fon instance uf ¡4 = (3(t), witit t trenscendental over ’, if (2 is the curve defluied b>’ a’ +y2 + 1 oven (3, and E = 11(C) is the fleid of rationa] functions on (2 oven 11, titen 11 1K (13(t), but E is not isomonpitic to 111(t) since E is not ordenable (a’2 + ~2 + 1 — O un E). Real Liirotit’s theorem states under which conditions E and 11(t) are isomorpitic’. More precisel>’: Real Liiroth’s Thearem (Field theor>’ version). Even>’ ordenable subfield E of the field (3(t) (t transcendental oven ¿U) transcendental over 11, la isomorpitic to 111(t). We remark itere titat it is equivalent, fon a subfield E of (3(t), to be botit orderable and stnictly containing IR. ¿md tobe orderable and transcendental oven 11. lix fact, if 1< stnictl>’ contains 11, titen eititer is contained un ¿i~ or contains some element in (13(t) \ (3. lix tite latter case, clearí>’ it is transcendental oven IR. lix tite first situation, E can not be orderable (since it will be an algebraic extension of 11). Tite converse is trivial. Equivaleixtí>’, fon algebraic curves, tite titeonem can be stated as follows: Real Lfiroth’s Theorem (Algebraic curves version). Ever>’ real rational plane curve can be parametnized oven tite reals. Let us remark that a real national plane cune (2 is a curve parametnizable over (3 ([2] Pp. 16,127,130), defined it>’ f E 11[a’,y], ¡ irreducible, and sucit titat 11(C) is onderable (titat la, (2 itas ininitel>’ man>’ real ‘This is equivalent to be R-isomorpbic. Wc. tbank the referee for pointing out this fact. A really eleznentary proof... 285 points). In otiter ivords, (2 is a true curve in 112 and its complexification is parametnizable oven ’ tite classical Lúroth’s theonem, titat the function fleld of the curve is isomorpitic to C(t) or 11(1), respectivel>’. Befone giving a proof of real Lñrotit’s theonem, we finst prove that botit statements are equivalent: let us assmne titat tite fleld titeor>’ version of real Liinoth’s titeonem itolds and let (2 be a real national plane curve. Titen 11(C) la orderable, and 11 ~ 11(C) ~ (3(t), witit 1 transcendental oven ’ version) one itas titat 11(C) is isomorpitie to 11(t), and titenefore (2 is panametrizable oven 11. Conversel>’, Jet us assume titat tite algebraic curves version of real Liirotit’s titeorem holds and let 1< be en ondenable subfield of (13(t) (1 transcendental oven (3) transcendental oven 11. Titen, &mce; tite tnanscendence degnee of 1< oven 11 is one, titene exists a curve (2 defined by aix irreducible ¡ E 11[a’, y] sucit titat E = 11(C). Fnrtitermore, (2 is a real rational plane curve (11(C) is orderable, and (2 is parainetnizable oven ’ proofs of tite titeorem can be deduced from [1]or [61(witere algonititmic tecitniques develop sorne ideas un [31). Tite appnoacit underl>’ing parametnization algonitbins can be applied to derive a dinect and constructive proof [5],[6]: un orden to panametnize a rational curve b>’ means of adjoint cunes, one considers the intensection of tite curve witit a linear subsystem, of dimension one, of a linean s>’stem of adjoint curves, obtained by iixtnoducing finitel>’ man>’ simple points on tite original curves as simple base points of tite linear system. Titenefore, since tite system of adjoint cunes can be computed witit ground field openations, it holds that tite rational curve can be parametnized oven tite field extension of tite gnound fleid where tite coondinates of tite simple points belong to. Titus, since aix>’ real curve itas infinitel>’ man>’ simple real points it follows titat, taking real simple points un tite sketched algonititm, en>’ real rational plane cune can be pananietrized oven tite reala; and titenefare a direct and constructive proof (since methods fon detenmining simple points of rational curves over optimal extensions are provided in [3],[6])of real Liirotit’s titeonem la denived. Afro, a dinect but 286 2’. Recio ¡md J. R. Sendra non constructive proof can be given using tite ideas of [1]: first one sbows titat imden tite h>’pothesis of tite titeorem, 1< is of genus zero (genus is defined ha [1] titrough divisors). Since by exteuiding tite base fleld from 11 to ’ transcendental extension of tite fleld of constents IR. But tite itypothesis titat 1< is orderable implies titat it itas a real place, namel>’, a place Witit IR as residue fleld, thus of degree one. 2 An Elementary Proof of Real Lúroth’s Theorem As anixonixced before, tite aim of this note la to provide an elementan>’ pnoof of titis titeonem. Uy elementary we mean that it does not use material beyond witat is stendand ha tite traditional presentation of tite classical Lñnotit’s theonem. Of course it requires tite concept of onderable fleld, or —in tite otiten version— of tite idea (quite natural) of real plane curve (2, un tite sense of being defined by a real polynomial and itaving en infinite nuniben of real points. Now, assuming titat (2 admits a national panametrization witit complex coefficients, we want to conclude titat it aleo itas a rational parametnization witit real coefficients. Jet ?Q) be a propen (i.e. an ahnost alWa3’s one to one) complex rational parametnization of (2, titen we will proceed as follo’ws: flrst oxte associates with (2 aix additional curve (2 titat provides tite complex panameten values titat generate —via 1’— tite real points on (2; aftenwards, one proves titat ¿2 itas one real component (2* titat is eititen a cincle or a lime, and finail>’ one shows titat ifM (mx(t), vn2(I)) la en>’ real panametrization of titis real component (2* of (2, titen ‘P(mi(t) +ivn2(t)) is a real parainetrization of (2. Titus, shace C~ is always parametnizable over 11, one concludes titat Cis panametnizable oven 11. More precisel>’, let 1 = 4 + it2, t1, ~2 E IR, denote a genenic complex A reafly elementary proof... 287 number. Then tite parametrization P(t) can be Wnitten un tite form: PQ) = 721(11,12) 722(11, 12) ivitere ~ it1, it2, ¡1, ¡2 E R¡a’,y]. Now, since (2 is real, titere exist infinitel>’ many poiuits (ti, 12) E 1112 sucit titat P(t1 + i 12) is a real point on (2. Titenefone, if Y1 is the set of zenos (ti, t2) E fl2 of tlie pol>’nomials ¡. E 11¡a’, yj, i = 1,2, tite curves ~1 ¿md ~2 itave haflnitely many common poiuits, and itence, the>’ have common components. Jet (2 be tite curve defined as tite union of tite common components of F~ auid ~2. It is a real curve, called tite associated curve witit (2 and P(t). lix the following, we enalyze the algebraic properties of (2. We start with tite followiuig tecitnical lemma. It nougitly means titat a curve xix qj 2 defined by a real polynomial, titat is intersected just on one point a pencil of trul>’ complex limes a’ = ay + 1 (Le. non real), must be eititer a conic or a lime. Tite second part of tite lemma, specifying tite kiuid of couiic is not really needed ha our proof, since we alvvays lcnow itow to panametnize a conic, but describes en hatenesting fact. Lemma 1. Leí ¡ E 11[a’, y] be a non constaní polysiomial, and a a non real complez number, sucit dial, for almoet allí É 3.4(ayy>(a’ — ay>1, 1=0 288 2’. Redo and J. R. Sendra