A really elementary proof of real Lüroth's theorem

ClassicalLifroth theoremstatesthat every subfield 1< of K(t), where t is a transcendental elementoverK, such that 1< stnictly containsK, mustbe 1< = K(h(t)), fon sornenon constantelement hQ) e K(t). Therefore,E la K-isornorphic to K(t). This result canbe provedwith elementaryalgebraictechniques, and therefore it la usual!>’ includedin basiccourseson fleld theoryor algebraic curves. In this papenwe 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 comnplexficíd), when doceit hoid that E is isonxorphicto R(t)? Obviously, a necesear>’ condition is that 1< admite an ordening. Herewe provethat this condition is also sufflcient, and we cali such statementthe Real Lúroth’s Theorenx. There are severalwaysof pnovingthis result (Riemann’stheorem,Hilbert-Hurwitz ¡3]), but we claim tbat proof is really elementary,since it don require just sornebasic backgroundas in the elassicalversionof Liiroth’s. 1 Real Lfiroth’s Theorem Liiroth’s Titeoremusuall>’ appearsin counseson fleld titeor>’ or in courses on algebraiccurves,and -~as it is well known— statestitat every subfleld *Partiafly supportedby CICyTPB92/0498/C0201(GeometríaReal y Algoritmos), Espnit/flra6846 (Poso),and TIC-1026-CE. ~Partiaflysupportedby Univ. Alcalá Proy. 030795. AMS subjectclassificatioxr 14H05,141>05. Servicio PublicacionesUniv. Complutense.Madrid, 1997. 284 2’. Recio¡md J. R. Sendra E of tite fleld K(t) (t transcendental oven K) transcendentaloven K (this means,in particular, titat E containsK), is isomorpitic to X(t), see[4]voL II Pp. 515 (i.eE hastite form K(it(t)), fon sorneh(t) E K(t)), or equivalentí>’, that tite fleid of rationalfunctions of ever>’ K-rational planecurveis isomorphicto K(t), see 1 7] vol. 1 pp. 9. Let 110W .1<’ be an algebraic extensionof K, and E a subfleld of K’(t) (t transcendental oven It) witich is transcendental oven K. Titen tite natural questionof auial>’zing uf E is isomorphicto K (t) anises. Ir particular, if K = 11 auid K’ = <U one ma>’ stud>’ if E is isomorpitie to 11(t). Cleaní>’, titis is not tnue fon ever>’ E. Fon instanceuf ¡4 = (3(t), witit t trenscendental over <U, it itolds titat 11 E (3(t), but E la not isomonpiticto 111(t). Similaní>’, if (2 is the curvedefluiedb>’ 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) sinceE is not ordenable(a’2 + ~2 + 1 — O un E). RealLiirotit’s theoremstatesunder which conditionsE and 11(t) are isomorpitic’. More precisel>’: Real Liiroth’s Thearem (Field theor>’ version). Even>’ ordenablesubfield 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 orderableand stnictly containing IR. ¿md tobe orderableand transcendental oven11. lix fact, if 1< stnictl>’ contains11, titen eititer is containedun ¿i~ or containssomeelementin (13(t) \ (3. lix tite latter case, clearí>’ it is transcendental oven IR. lix tite first situation,E cannot be orderable(sinceit will be an algebraicextensionof 11). Tite converseis trivial. Equivaleixtí>’, fon algebraiccurves, tite titeonemcanbe statedas follows: Real Lfiroth’s Theorem (Algebraic curvesversion). Ever>’ real rational planecurve can be parametnizedoven tite reals. Let usremarkthat a realnationalplanecune(2 is a curve parametnizableover (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 equivalentto beR-isomorpbic. Wc.tbanktherefereefor pointingout this fact. A really eleznentaryproof... 285 points). In otiter ivords, (2 is a true curve in 112 and its complexification is parametnizableoven <U, i.e. it is tite Zariski closureof a non constant rational map from (1) to <¡9. A similar definition gives tite concept of curveparametnizableoven tite neals. Botit imp1~r, b>’ 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 statementsare equivalent: let us assmnetitat 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 <U. Titus, by realLiirotit’s theorem(field titeon>’ version) one itas titat 11(C) is isomorpitieto 11(t), andtitenefore(2 is panametrizableoven11. Conversel>’,Jet us assumetitat tite algebraiccurvesversion of real Liirotit’s titeoremholds and let 1< be en ondenablesubfield of (13(t) (1 transcendental oven (3) transcendental oven 11. Titen, &mce tite tnanscendence degneeof 1< oven 11 is one, titeneexistsacurve (2 defined by aix irreducible¡ E 11[a’, y] sucit titat E = 11(C). Fnrtitermore,(2 is a real rationalplanecurve (11(C) is orderable,and (2 is parainetnizable oven <U). Titenefore,applying realLiinotit’s theoremfon algebnaiccurves, oneobtainsthat (2 is parametnizableoven 11. Ir tite sequel,wefocuson tite algebraiccurvesversionof realLiinotit’s titeonem. Direct, nonelementar>’proofs of tite titeoremcanbededuced from [1] or [61(witere algonititmic tecitniquesdevelopsorneideasun [31). Tite appnoacitunderl>’ing parametnizationalgonitbinscan be appliedto derive adinect andconstructiveproof [5],[6]: un orden to panametnizea rationalcurveb>’ meansof adjoint cunes,oneconsidersthe intensection of tite curvewitit alinearsubsystem,of dimensionone,of alineans>’stem of adjoint curves,obtainedby iixtnoducingfinitel>’ man>’ simplepoints on tite original curvesassimple basepoints of tite linear system.Titenefore, since tite systemof adjoint cunescan be computedwitit groundfield openations,it holdsthat titerational curvecanbe parametnizedoventite field extensionof tite gnoundfleid where tite coondinatesof 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 rationalplanecunecan be pananietrizedoven tite reala;and titenefarea direct andconstructiveproof (since methodsfon detenminingsimplepointsof rationalcurvesover optimal extensionsare providedin [3],[6]) of realLiirotit’s titeonemla denived.Afro, adinectbut 286 2’. Recio ¡md J. R. Sendra nonconstructiveproofcanbegiven usingtite ideasof [1]: first onesbows titat imdentite h>’pothesisof tite titeorem,1< is of genuszero (genus is definedha [1] titrough divisors). Since by exteuidingtite basefleld from 11 to <U, the genusof IR auidof the result, IR’, of sucit extension,is kept tite same([1], p. 99), we must jnst prove that tite extendedsubfield 1=’of (3(t) is of genuszero. Titis is acitievedusing tite classicalLúroth theonemandtite fact ([1] p. 22) titat (3(t) la of genuszero. Now [1], p. 22, sitows titat for genuszeroalgebraicfuixetion fields IR, if titere is at least one place of degreeone, titen 1K is a purel>’ transcendental extensionof 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 anixonixcedbefore, tite aim of this note la to provide an elementan>’ pnoof of titis titeonem. Uy elementary we mean that it does not use material beyondwitat is stendandha tite traditional presentationof tite classicalLñnotit’s theonem.Of courseit requirestite conceptof onderable fleld, or —in tite otiten version—of tite idea (quite natural) of real plane curve (2, un titesenseof beingdefinedby arealpolynomial and itavingen infinite nunibenof realpoints. Now, assumingtitat (2 admits a national panametrizationwitit complexcoefficients, we want to concludetitat it aleo itas a rationalparametnizationwitit real coefficients. Jet ?Q) bea propen(i.e. an ahnostalWa3’s oneto one)complexrationalparametnization of (2, titen we will proceedas follo’ws: flrst oxte associateswith (2 aix additional curve (2 titat providestite complexpanametenvaluestitat generate—via 1’— tite realpoints on (2; aftenwards,oneprovestitat ¿2 itas one realcomponent(2* titat is eititen a cincle or a lime, and finail>’ one showstitat ifM (mx(t), vn2(I)) la en>’ realpanametrizationof titis real component(2* of (2, titen ‘P(mi(t) +ivn2(t)) is a realparainetrizationof (2. Titus, shaceC~ is alwaysparametnizableover 11, oneconcludestitat Cis panametnizableoven11. More precisel>’,let 1 = 4 + it2, t1, ~2 E IR, denotea geneniccomplex A reafly elementaryproof... 287 number. Thentite parametrizationP(t) canbe 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 arealpoint 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 itavehaflnitely many common poiuits, and itence, the>’ havecommon components.Jet (2 be tite curve defined as tite union of tite common componentsof F~ auid ~2. It is a real curve, calledtite associatedcurve witit (2 andP(t). lix the following, we enalyzethe algebraicpropertiesof (2. We start with tite followiuig tecitnical lemma. It nougitly meanstitat a curve xix qj 2 definedby a real polynomial, titat is intersectedjust on onepoint a pencil of trul>’ complex limes a’ = ay + 1 (Le. non real), must be eititer a conic or a lime. Tite secondpart of tite lemma, specifying tite kiuid of couiic is not really neededha our proof, since we alvvays lcnow itow to panametnizea conic, but describesen hatenestingfact. Lemma 1. Leí ¡ E 11[a’, y] be a non constanípolysiomial, and a a non real compleznumber,sucitdial, for almoetallí É <U~ oneitasdeg~(f(ay+ t,y)) = 1. Titen it holde thai ¡(a’,y) definescititer a line or a conte. Partitermore, iff itas deyreeIwo, titen: (1) 1ff la reducibleove