段和有负平方的定向域（可编辑）

段和有负平方的定向域（可编辑） 段和有负平方的定向域 h ////0>. //.段和有负平方的定向域 1 2 W. R ump ,杨义川 1 Univ ersit?t Stuttgart, F ac h b ereic h Mathematik, Institut für Algebra und Zahlen theorie, Pfa?en w aldring 57 D-70550 Stuttgart, German y 2 北京航空航天大学数学与系统科学学院数学系,数学、信息与行为教育部重点实验室 北京 100191 摘要: 本文中, 我们研究了段和有负平方的定向域间的关系对于线性序域 K 和代数元 i 满足 2 i ?1 , 我们在用 K 的有理段参数化了 Ki 上的一类格序群结构的理论基础上,建立了域 Ki 上的定向序与和 K 的乘性段 V 间的对应关系关键词: 定向域,负平方,格序,段 中图分类号: O152.6 Segmen ts and directed partially ordered ?elds with negativ e squares 1 2 R UMP W olfgang , Y ANG Yic h uan 1 Univ ersit?t Stuttgart, F ac h b ereic h Mathematik, Institut für Algebra und Zahlen theorie, Pfa?en w aldring 57 D-70550 Stuttgart, German y 2 Sc ho ol of Mathematics and System Sciences, LMIB of the Ministry of Education, Beijing Univ ersit y of A eronautics and Astronautics, 100191 Be ijing, China Abstract: In this pap er, w e study the relationship b et w een segmen ts and directed orders on the ?eld with negativ e squares. F or a linearly ordered ?eld K , w e parameterize a class of 2 lattice-ordered group structures on Ki with i ?1 b y rational segmen ts of KThen, w e establish a corresp ondence b et w een directed orderings on the ?eld Ki and m ultiplicativ e segmen ts V of KKey w ords: directed ?eld, negativ e square, lattice order, segmen t 0 In tro duction Artin and Sc hreier [1], Johnson [2], F uc hs [3], and man y others for instance, Szele [4], Szigeti [5], and Serre [6] etc. observ ed the non-existence of a compatible total order on rings with negativ e squares. On the other hand, Bourbaki [7], Birkho? and Pierce [8] etc. pro v ed that 基 金 项 目: Supp orted b y the Researc h F und for the Do ctoral Program of Higher Education Gran t 20091102120045, Beijing Municipal Natural Science F oundation Gran t 1102027 and NSF C Gran t 11271040 作 者 简 介: R UMP W olfgnag , male , Professor, ma jor researc h direction : Algebra. Corresp ondence author : Y ang Yic h uan , male , Professor , ma jor researc h direction : algebra. - 1 -h ////. //. lattice-ordered ?eld in whic h an y square is p ositiv e m ust b e totally ordered, and this result has b een extended to sk ew ?elds b y Y ang in [9]. Sc h w artz [10] sho w ed that the ?eld of algebraic n um b ers admits no partial order with resp ect to whic h it is a lattice-ordered ?eld. Y ang [11] guran teed the existence of directed partially ordered ?elds with negativ e squares. C on tin uing in this direction, Sc h w artz and Y ang [12] sho w ed that almost all ?elds of c haracteristic 0 , including the ?eld of complex n um b ers, can b e made in to a directed partially ordered ?eld. In this pap er, 2 w e c haracterize the directed orders on the ?eld Ki with i ?1 for a linearly ordered ?eld K b y segmen ts. W e ?rst exhibit a particular class of l -group structures on Ki and sho w that they can b e parameterized b y p ositiv e elemen ts of K Theorem 2. As a simple consequence, w e pro v e that the directed ordered ?elds with negativ e squares constructed in [11] need not b e lattice-ordered see Remark 1. Then, w e establish a corresp ondence b et w een directed orderings 1 ?1 of the ?eldKi and segmen tsV ofK whic h satisfy the implication 0 a? / Va?a ? / V 2 Theorem 3. Finally , w e c haracterize these V as con v ex additiv e subgroups with the prop ert y × ?1 that for a? K , either a or a b elongs to V Theorem 4. Note that the latter prop ert y holds for an y v aluation domain with quotien t ?eld K1 Preliminaries A p artial ly or der e d gr oup p o-group for short G G;+;? is a not necessarily ab elian group G;+ with binary op eration + and iden tit y elemen t 0 , where the in v erse of a mem b era ofG is denoted b y?a and a partially ordered set G;? in whic ha ? b implies thata+c? b+c and c+a? c+b for all a; b; and c in GAn elemen t a of G is positive if a? 0The p ositiv e ?0 0 cone and strictly p ositiv e cone of G , denoted b y G and G , is the set of all p ositiv e elemen ts and all strictly p ositiv e elemen ts of G resp ectiv elyF or t w o elemen ts a;b in a p o-group, w e use a ‖ b to denote that a and b are incomparable. A p o-group is Archimedean if na ? b for all in tegers n implies a 0A p o-group G;+;? is called a dir e cte d gr oup if the partial order ? is a directed order, i. e., for an y a;b ? G , there exists c ? G suc h that c ? a and c ? bNote that Cli?ord pro v ed that a p o-group G;+;? is a directed group if a nd only if G can b e ?0 generated b y GA directed group G;+;? is called a lattic e-or der e d gr oup l-group for short if the partial order ? is a lattice order, that is, if eac h pair of elemen ts x and y in G ha v e a unique least upp er b oundx?y and a unique greatest lo w er b oundx?yAn l-groupG is a linearly or totally ordered group if the order is a simple order: x? y or y? x in G for all x;y? GIf G;+;? is + an l-group and a b elongs to G , then the p ositiv e part of a is a a?0 , the negativ e part of a+is a ?a?0 , and the absolute v alue of a is|a| a??aIt is easily seen that a a ?a +and that|a| a +aA directed ring R R;+;?;? is a ring that is partially ordered and has the follo wing - 2 -h ////. //.rop erties: i R;+;? is a directed ab elian group, and ii a? 0 and b? 0 imply that ab? 0A directed ring R is Arc himedean if the directed additiv e group R;+;? is Arc himedean. Lattice-ordered rings and totally ordered rings can b e de?ned similarlyA directed ?eld is a directed ring whose underlying ring is a ?eld. A partially ordered ring R;+;?;? has a ne gative 2 squar e if there is an elemen t r? R with r ? 0F or basic theory of partially ordered algebraic systems, w e refer the reader to [13]. 2 Segmen ts and directed orders on the additiv e group of Ki 2 Let K;? b e a linearly ordered ?eld and i ?1A prop er subset V of K will b e called a segment of K , if V satis?es 1 0? V , and 2|a|?|b|; b? V implies a? VF or an y segmen t V of K , de?ne ?1 P :a+bi? Ki| a;b? 0;b 0? a 0;ab ? / V: V Let A;B b e t w o subsets of a additiv e group, w e use?A , A+B and A?B to denote the set?x| x? A ,a+b| a? A;b? B , and A+?B , resp ectiv elyLemma 1. The P de?ne d ab ove satis?es: V a P ??P 0 ; V V b P +PP ; V V V c P ?P KiV V Pr o of. a T rivial. b Assume that a+bi;c+di? PThen a;b;c;d? 0 , without loss of generalit y , supp ose V a+c a that b 0 , whic h implies a 0 and a/b ? / V , so a +c 0If d 0 , then ? ? / V , b+d b a+c a c whic h implies ? / VThere is no loss of generalit y in assuming that ?It follo ws that b+d b d a a+c c a+c ? ? , and th us ? / Vb b+d d b+d c F or all a +bi ? KiCho ose y 0 and y bThere exists z 0 with z ? K\V since V ? KCho ose x 0 , x a , x? yz and x? a + y?bz since K is totally ordered. x x?a Then x?a;y?b 0; ? z and ? z note that a ?eld is totally ordered if and only if an y y y?b ?1 x x?a square is p ositiv e, it follo ws that y? 0 implies y ? 0 , hence ; ? / VFinally , it is clear y y?b that a+bi x+yi? x?a+y?bi ?0 F or a partially ordered algebraic system S;? , w e use S to denote the p ositiv e cone. By the theorem of Cli?ord and Lemma 1 it follo ws that an y segmen t determines a compatible - 3 -h ////. //.artial order for whic h the additiv e group Ki is a directed additiv e group. F urthermore, w e ha v e Theorem 1. V 7? P de?nes a bije ction b etwe en se gments V of K and p artial or derson V ?0 ?0 Ki such that KKi and Ki is a dir e cte d additive gr oup with 1 0bia for al l a? K and 0 b? K ; 2 ther e exists a 0 in K with ?a? iPr o of. By Lemma 1 the p ositiv e cone P de?nes a directed order on Ki via V ? P for ; ? KiSupp ose that bia ? K for some 0 b ? K and a , then V a?bi ? P , a con tradiction, whic h implies 1 since 0 bi is ob vious. V ? K implies that V ?0 there is an a? K \V , so a+i? P whic h implies 2. V Con v ersely , if Ki;+;? is a directed group with the prop erties ab o v e. De?ne ?1 V :ab ? K | 0??|a|+|b|i: By 2 there exists 0 a ? K with 0a +i , whic h implies V ? KThe condition 0 i implies 0? VF urthermore,|a|?|b| and b? V implies 0??|b|+i and a ? V since 0?|a|+i is imp ossible b y|a|+i?|b|+iSo V is a segmen t. W e next to sho w that the p ositiv e cone of Ki is P , that is, V ?1 0? a+bi? a;b? 0;b 0? a 0;ab ? / V : “? ”: Assume b 0Then ?bia , whic h is a con tradiction and th us b ? 0Supp ose a 0 , so 0??a? bi , whic h con tradicts 1, whence a? 0If b 0 , then a 0 , ?1 and ab ? / V?0 ?0 “? ”: F or b 0 and a? 0 , w e get 0? a a+bi since KKiF or b 0 , w e ha v e ?1 a 0 and ab ? / V whic h implies 0?|a|+|b|i a+bi?1 Finally , it is trivial that V ab ? K ||a|+|b|i? / P V 0 W e call a segmen t V rational if it has the form V :a? K ||a| for some ? K Theorem 2. The or der in The or em 1 makes Ki into an l -gr oup if and only if V is r ational. Pr o of. Assume that V V with 0 ? KT o pro v e 0? a + bi ? Ki for all a+bi? KiF or b 0 it is trivial. Case I. b 0 : Then 0?a+bi c+bi with c a;b Case I I. b 0 : Then 0?a+bi 0;a?b Con v ersely , supp ose that Ki is an l -group, but V is not rational. Let 0?i a+biThen ?1 ?1 b? 1 , a 0 and ab ? / VTh us there exists ? / V with 0 ab : Ho w ev er, b +bi? 0 b and b +bi ? i are plain for b 1F urthermore, if b 1 , then ? ? / V whic h implies b?1 b +bi? iTherefore, a+bi?b +bi a?b 0 , a con tradiction. - 4 -h ////. //.emark 1. By Theorem 2 it follo ws that the directed ?elds with negativ e squares constructed in [11] need not b e lattice-ordered: Let F b e a totally ordered ?eld, K b e the quotien t ?eld of the p olynomial ring F[x] , and 2 let i b e a solution of x + 1 0It is w ell kno wn that F[x] is a totally ordered ring with n resp ect to the order: a +a x +??? +a x 0 if and only if a 0 in F , and K is a totally 0 1 n n fx ordered ?eld with resp ect to the order 0 if and only if fxgx 0 in F[x]Let v b e gx the negativ e v aluation on K de?ned b y the di?erence of the degree function on KThen the set ?0 P a+bi? Ki| a;b? K ; and if b? 0; then va vb will b e the p ositiv e cone of a partial orderon Ki for whic h Ki is a directed ?eld see Corollary 2.3 in [11]. F urthermore, it is straigh tforw ard to v erify that the directed order de?ned b y P satis?es the conditions of ab o v e Theorem 1. No w w e sho w that the corresp onding segmen t ?1 V ab | 0??|a|+|b|i? V a? K ||a| 0 0 for an y elemen t ? KAssume that there exists ? K with V VIt is clear that ?0 FV , so v 0 and V 1? Ho w ev er, v 1 v and 0? 1 +i , a con tradiction. W e also note that Remark 2.5 in [12] pro v ed that the directed ordered in [11] is not a lattice b y viewing the p olynomial rings as a v ector space. 3 Segmen ts and directed orders on the ?eld Ki ?0 ?0 Lemma 2. A se gment V of K de?nes a p artial or der such that KKi and Ki is a dir e cte d ?eld if and only if i 1? V , and ab?1 ii a;b? / V;a 0;b 0 implies ? / Va+b Pr o of. If Ki is a directed ?eld, that is, P PPAssume 1 ? / V , then 1 +i ? P V V V V 2 implies 1 + i 2i ? P ; in con tradiction with Theorem 1 1 and th us i holds. Let V 0 a;b? K \V , then a+i;b+i? P and hence a+ib+i ab?1+a+bi? P , whic h V V ab?1 implies ? / Va+b ?0 Con v ersely , supp ose that i and ii hold. Let a+bi;c+di? PThen a;b;c;d? KV Case I: bd 0Then ac? bd ? 0Without loss of generalit y , assume that d 0If ?1 ad + bc 0 , then b;c 0 whic h implies a 0 and ab ? / VSo ac? bd ac 0 and ac?bd ac a ? / Vad+bc bc b ?1 ?1 ?1 ?1 Case I I: bd 0 implies that a;c 0 and ab ;cd ? / VTh us ab ;cd 1 since 1? V , a c ?1 ac?bd b d and hence ac?bd 0ii implies that ? / V , whic h implies ? / V a c + ad+bc b d - 5 -h ////. //.xample 1. The order constructed in the pro of of the main theorem in [11] can b e de?ned b y a segmen t. By the condition i of Lemma 2 and the de?nition of the segmen t w e ha v e follo wing in teresting corollaryCorollary 1. If 1 is a str ong unit in K , then Ki c annot b e p artial ly or der e d such that Ki is a dir e cte d ?eld for any se gment V of KEsp ecially , if K is an l -sub?eld of the Arc himedean totally ordered ?eldR , then the corol- lary ab o v e is applicable. Equiv alen tly , the ?eld K in Theorem 3 can b e considered as a non- arc himedean linearly ordered one. Let us call a segmen t VK multiplicative if it satis?es 1 ?1 iii 0 a? / V implies a?a ? / V2 Theorem 3. V 7? P de?nes a bije ction b etwe en multiplic ative se gmentsV and p artial ly or ders V 0 ? on Ki such that Ki is a dir e cte d ?eld with bi?? a for al l a? K , b? K , a nd k i for some k? K2 a ?1 1 ?1 Pr o of. ii? iii : Set a b , then a?a ? / V2a 2 iiiii : Without loss of generalit y , supp ose that 0 a bW e w an t to sho w that 1 ?1 ab?1 2 2 a?a ? , whic h is equiv alen t to sho w ba +1? aa +12 a+b 1 ?1 iii? i : Supp ose that 1? / V , so 1?1 ? / V , a con tradiction. 2 2 The conditions 1 and 2 in Theorem 1 are simpli?ed: If i? 0 , then i ?1? 0 , whic h is imp ossible. ′ 1 ?1 Remark 2. Let a a?a for a 0Then iii is equiv alen t to 2 ′ 0 a? / V0 a ? / V: ?1 ′ In fact, 0 a ? / V implies that a 1 and th us a a , whic h implies a 0: F urthermore, ′ 2 for a 0 w e ha v e i ?a? i ?a since i +a 0 giv es that i +a 0 whic h giv es that 2 1 ?1 2ai 1?a , and th us i ? a?a 2 Corollary 2. If Ki is an l -?eld with K as a line arly or der e d sub?eld. Then i ‖ a for al l a? K , or ther e exist a;b ? K such that a i b , or for al l a? K ther e exists b? K such that a?i? bPr o of. It su?ces to pro v e that ev ery directed order in Theorem 3 cannot mak e Ki in to ′ ′ an l -?eld. Assume that V VThen ? / V and 1 , and th us 0 ; so ? VIn con tradiction with the de?nition of a m ultiplicativ e segmen t. - 6 -h ////. //.emark 3. If on the ?eldC there exists a partial order suc h thatC is an l -?eld with resp ect 1 ?1 a to Theorem 3, then there exists 0 a ? R with i aTh us i a? a ; whic h 2 2 a implies i for all n ? N b y induction. Ho w ev er, w e cannot get i ? 0 in general, whic h n 2 implies that the order cannot b e Arc himedean b y a simple argumen t. Note that all lattice orders constructed in Wilson [14] are Arc himedean. Theorem 4. A se gment V of K is multiplic ative if and only if it satis?es the fol lowing c onditions: V is an additive sub gr oup; V is c onvex; ?1 ?a? K\0 : a? V or a ? Va+b Pr o of. If V is a m ultiplicativ e segmen t and 0 a b in V , then a b implies 2 a+b a+b 1 ?1 ? VAssume that a +b ? / VThen ? a +b? a +b ? / V , a con tradiction. 2 2 2 Th us a+b? VNo w let a;b? V b e arbitraryThen|a|+|b|? V and|a+b|?|a|+|b| implies a+b? VHence is pro v ed. is clear. 1 |a||a| ?1 ?1 ?1 : If? 0 and a;a ? / VThen|a|;|a| ? / V implies a con tradiction: 0 ? / V 1 |a|+|a| Con v ersely , supp ose that V satis?es , and . Let 0 a ? / VBy it follo ws ?1 1 ?1 ?1 ?1 that a ? VAssume that a?a?1? VThen a?a ? V and a a?a ++a ? V , 2 a con tradiction. Corollary 3. Every c onvex valuation sub domainV ? K in its quotient ?eldK non-ar chime de an line arly or der e d is a multiplic ative se gment. Pr o of. The conditions , and are clearly satis?ed. By Corollary 1 1 is not a strong unit, whic h implies that K is non-arc himedean. Note that a m ultiplicativ e segmen t need not come from a real v aluation ring, since there is a sea of non-arc himedean linearly ordered ?elds. Example 2. Let K b e the surreal n um b er ?eld. Then K is a nonarc himedean linearly or- dered ?eld [15]. Then ev ery con v ex v aluation sub domain V ? K in its quotien t ?eld K is a m ultiplicativ e segmen t. Remark 4. The condition in Theorem 4 can b e substituted b y 1? V: ?1 A ctually , implies 1 1 ? VCon v ersely , assume thata? / V , then|a|? / V implies|a| 1 , ?1 1 1 and th us 1 |a| | | 0 , whic h implies ? V: a a - 7 -