EJNIKO METSOBIO POLUTEQNEIO. Oi p-adikoð arijmoð kai mða efarmog stic Diofantikèc exis seic

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1 EJNIKO METSOBIO POLUTEQNEIO SQOLH EFARMOSMENWN MAJHMATIKWN KAI FUSIKWN EPISTHMWN TOMEAS MAJHMATIKWN Oi p-adikoð arijmoð kai mða efarmog stic Diofantikèc exis seic DIPLWMATIKH ERGASIA thc EugenÐac A. AlexopoÔlou Epiblèpousa: SofÐa LampropoÔlou Kajhg tria E.M.P. Aj na, IoÔnioc 2007

2 EJNIKO METSOBIO POLUTEQNEIO SQOLH EFARMOSMENWN MAJHMATIKWN KAI FUSIKWN EPISTHMWN TOMEAS MAJHMATIKWN Oi p-adikoð arijmoð kai mða efarmog stic Diofantikèc exis seic DIPLWMATIKH ERGASIA thc EugenÐac A. AlexopoÔlou Epiblèpousa: SofÐa LampropoÔlou Kajhg tria E.M.P. EgkrÐjhke apì thn trimel exetastik epitrop Eugènioc Aggelìpouloc SpÔroc Argurìc SofÐa LampropoÔlou Kajhght c E.M.P. Kajhght c E.M.P. An. Kajhg tria E.M.P. Aj na, IoÔnioc 2007.

3 Prìlogoc Sthn paroôsa diplwmatik ergasða melet ntai oi p-adikoð arijmoð kai parousiˆzetai mða efarmog touc stic Diofantikèc exis seic. Pio sugkekrimèna, sto pr to kefˆlaio gðnetai mða istorik anadrom, sthn opoða faðnetai o arqikìc orismìc twn p-adik n arijm n wc to sônolo twn apì aristerˆ peperasmènwn ajroismˆtwn dunˆmewn enìc pr tou p, kaj c kai ta kðnhtra pou od ghsan sthn kataskeu touc. kˆpoiwn Diofantik n exis sewn. Autˆ tan h eôresh akeraðwn lôsewn Sto deôtero kefˆlaio eisˆgetai h ènnoia thc mh arqim deiac nìrmac kai parousiˆzontai oi kuriìterec idiìthtec thc, ìpwc to ìti se èna q ro me mh arqim deia nìrma kˆje trðgwno eðnai isoskelèc to ìti kˆje shmeðo miac mpˆlac eðnai kèntro thc mpˆlac. EpÐshc, orðzetai h p-adik nìrma pˆnw apì to s ma twn rht n arijm n, h opoða eðnai mh arqim deia. Ostrowski (bl. SÔmfwna me to Je rhma Je rhma 5) kˆje nìrma pou mporeð na oristeð pˆnw sto QI eðnai isodônamh eðte me th sun jh apìluth tim, eðte me kˆpoia p-adik. Sto trðto kefˆlaio kataskeuˆzetai to s ma twn p-adik n arijm n QI p me mejìdouc anˆlushc kaj c kai me algebrikèc mejìdouc, kai melet ntai oi basikèc tou idiìthtec, oi opoðec xeqwrðzoun gia thn aplìthtˆ touc se sqèsh me thn klassik perðptwsh twn pragmatik n arijm n me th sun jh apìluth tim. Gia parˆdeigma, mða akoloujða p-adik n arijm n (a n ) eðnai Cauchy an kai mìno an lim n a n+1 a n p = 0. EpÐshc, eðnai polô eôkolo na brðskoume sto QI p rðzec poluwnômwn, ìtan autˆ èqoun rðzec modulo p. Autì mac lèei to L mma tou Hensel, to opoðo diatup noume sto tètarto kefˆlaio (bl. Je rhma 11). Sto tètarto kefˆlaio, parousiˆzetai epðshc mða shmantik efarmog twn p- adik n arijm n sthn epðlush Diofantik n exis sewn pou eðnai tetragwnikèc morfèc n metablht n. H epðlush basðzetai sto l mma tou Hensel kai sthn Topik -Olik Arq, pou lèei ìti: mða Diofantik exðswsh èqei lôseic sto QI an kai mìno an èqei lôseic se kˆje QI p kai sto IR. H Topik -Olik Arq den isqôei pˆnta, allˆ isqôei sthn perðptwsh twn tetragwnik n morf n. Autì eðnai to Je rhma Hasse-Minkowski, tou opoðou kai parajètoume thn apìdeixh. 1

4 Tèloc, sto pèmpto kefˆlaio, parajètoume kˆpoiec ˆllec efarmogèc twn p-adik n sth Fusik, th BiologÐa kai th JewrÐa Plhrofori n. KleÐnontac aut thn eisagwg ja jela na epishmˆnw th diaforetikìthta, wc proc ton trìpo skèyhc, pou dièpei ta majhmatikˆ twn p-adik n arijm n se antðjesh me ta klassikˆ majhmatikˆ. Aut h diaforetikìthta up rxe gia mèna mða prìklhsh kai mða euqˆristh èkplhxh, kaj ìti mìno me ta teleutaða eðqa epaf katˆ th diˆrkeia twn spoud n mou. Ja jela loipìn na euqarist sw idiaðtera thn epiblèpousa kajhg tria SofÐa LampropoÔlou gia to ìmorfo jèma pou mou prìteine, kaj c kai gia thn polôtimh bo jeiˆ thc kaj ìlh th diˆrkeia thc ekpìnhshc thc diplwmatik c mou ergasðac. 2

5 Introduction The subject of this diploma thesis is the field of p-adic numbers QI p and its use in solving Diophantine equations. We first present the motivation for the invention of p-adic numbers. In the next chapter we introduce the concept of a non-archimedean norm and its corresponding valuation and define the non-archimedean p-adic norm p, where p is a prime number. We also study the properties of a non-archimedean norm on a field. In chapter 3 we construct the field of p-adic numbers QI p, both analytically and algebraically. Analytically seen, the field QI p is the completion of the rational numbers QI with respect to the p-adic norm p. Algebraically, after constructing the ring of p-adic integers Z p as the inverse limit of the inverse system ( Z/p n Z, θ n m), where θ n m(a) = a (mod p m ), one may construct QI p as the field of fractions of Z p, Z p [1/p]. Studying Analysis in QI p turns out to be very interesting. Some of the most important results presented are the following: Both Z p and QI p are complete and Z p is the completion of the integers Z, with respect to the p-adic norm. Moreover Z p is compact, whereas QI p is locally compact. QI p is a totally disconnected Hausdorff topological space. QI p is not an ordered field and it is not algebraically closed. A sequence of p-adic numbers (a n ) is a Cauchy sequence if and only if lim n a n+1 a n p = 0. A series of p-adic numbers n=1 a n converges, if and only if lim n a n p = 0. Combining some of the above results with the algebraic structure of QI p we obtain Hensel s Lemma, a method for approximating the roots of a polynomial within finite time. Hensel s Lemma states that a p-adic integer root of a polynomial with p-adic integer coefficients exists if and only if there exists a simple root modulo p. We present some simple applications of Hensel s 3

6 Lemma, like determining the roots of unity and the squares in QI p, and deciding when a quadratic form has a p-adic solution. Furthermore, in chapter 4, we analyse the importance of Hensel s Lemma in solving Diophantine equations. More precisely, it can be combined with the Local-Global Principle, which states that an equation can be solved over QI if and only if it can be solved over all the QI p and IR. We note that the Local-Global Principle does not hold for any equation. But given the Local-Global Principle and having a method for finding p-adic solutions of an equation, rational roots can also be detected. For example, in the case of quadratic forms the Local-Global Principle is successful, as the Hasse- Minkowski Theorem states. Finally, in chapter 5, we describe briefly some of the applications of the fields of p-adic numbers in Physics, in Information Theory and in Biology. 4

7 Perieqìmena 1 MÐa sôntomh istorik anadrom H analogða tou Hensel Exis seic modulo p n Diofantikèc Exis seic Mh arqim deiec nìrmec pˆnw se s ma IK Nìrmec pˆnw se s ma IK 'Ena parˆdeigma: h p-adik nìrma sto QI Oi diaforetikèc nìrmec sto QI TopologÐa se s ma me mh arqim deia nìrma TopologÐa sto QI me thn p-adik nìrma 'Algebra se s ma me mh arqim deia nìrma 'Algebra sto QI me thn p-adik nìrma To s ma twn p-adik n arijm n H analutik kataskeu tou QI p O daktôlioc ektðmhshc tou QI p H algebrik kataskeu tou QI p AntÐstrofa sust mata kai antðstrofa ìria H kataskeu twn p-adik n akeraðwn Z p O daktôlioc Z p Anˆlush sto Z p ApeikonÐseic twn p-adik n arijm n Anˆlush sto QI p SÔnoyh kai sugkrðseic

8 4 To L mma tou Hensel kai mða efarmog stic Diofantikèc exis seic To L mma tou Hensel 'Allec morfèc tou L mmatoc tou Hensel Efarmogèc tou L mmatoc tou Hensel Topik kai Olik Arq To Je rhma Hasse-Minkowski 'Allec efarmogèc twn p-adik n arijm n 146 6

9 Kefˆlaio 1 MÐa sôntomh istorik anadrom Katˆ th diˆrkeia tou perasmènou ai na oi p-adikoð arijmoð kai h p-adik anˆlush èqoun anaptuqjeð idiaðtera kai plèon èqoun kentrikì rìlo sth jewrða arijm n. Autì sumbaðnei kurðwc gia dôo lìgouc: thn eukolða pou mac parèqei h gl ssa twn p-adik n arijm n sto na ekfrˆsoume sqèseic isotimðac metaxô akeraðwn kai thn aploôsteush thc melèthc twn rht n apì th skopiˆ thc anˆlushc. H idèa eðnai na eisaqjeð ènac nèoc trìpoc mètrhshc thc apìstashc metaxô rht n, mða nèa metrik, kai proèkuye apì sugkekrimèna probl mata thc JewrÐac Arijm n kai thc 'Algebrac. Den upˆrqei lìgoc na jewroôme monadik dunatìthta gia touc rhtoôc na èqoun th sun jh metrik. Gia parˆdeigma, mia opoiad pote ˆllh sunˆrthsh pou antistoiqðzei se kˆje zeôgoc rht n è- nan trðto kai ikanopoieð ton orismì thc nìrmac ja tan Ðswc exðsou kal gia th melèth touc. An xekin soume apì th sun jh nìrma kai th metrik pou aut epˆgei sto QI, kai pˆroume thn pl rwsh tou QI prosjètontac ta ìria twn akolouji n Cauchy rht n arijm n, katal goume sto s ma twn pragmatik n arijm n IR. An dialèxoume diaforetik nìrma, ja katal xoume se kˆti ˆllo. Autì akrib c eðnai kai to antikeðmeno thc melèthc mac. O pr toc majhmatikìc pou eis gage touc p-adikoôc arijmoôc tan o Kurt Hensel sta 1897, an kai o E. Kummer qrhsimopoioôse dh tic p-adikèc mejìdouc apì to O Kummer kratoôse kaj' ìlh th diˆrkeia thc zw c 7

10 tou sqèsh di' allhlografðac me to majht tou Kronecker, o opoðoc ègraye kai th diplwmatik tou ergasða pˆnw se aut n thn kateôjunsh. O Kronecker tan me th seirˆ tou kajhght c tou Hensel. O Hensel ìqi mìno spoôdase me touc Kronecker kai Kummer alla tan epðshc majht c tou Weierstrass kai xere kalˆ ton orismì twn pragmatik n arijm n apì ton Cantor kai tic idèec twn Weber kai Detekind gia thn analogða metaxu swmˆtwn arijm n kai swmˆtwn sunart sewn. Parakˆtw ja doôme giatð akrib c ènac majhmatikìc pou gn rize ìla autˆ ta ergaleða mpìrese na eisˆgei tic ènnoiec kai touc sumbolismoôc twn p-adik n arijm n kai twn p-adik n mejìdwn. 1.1 H analogða tou Hensel To kðnhtro tou Hensel tan kurðwc h analogða metaxô tou daktulðou Z twn akeraðwn, me to s ma-phlðko tou, touc rhtoôc QI, kai tou daktulðou CI[x] twn poluwnômwn me migadikoôc suntelestec me to s ma-phlðko touc CI(x). gðnoume pio sugkekrimènoi: èna stoiqeðo f(x) CI(x) eðnai mða rht sunˆrthsh, dhl. to phlðko dôo poluwnômwn p(x), q(x) CI[x] me q(x) 0: f(x) = p(x) q(x). AntÐstoiqa, ènac rhtìc arijmìc x QI eðnai to phlðko dôo akeraðwn a, b Z Ac me b 0: x = a b. EpÐshc, oi idiìthtec twn dôo daktulðwn eðnai arketˆ parìmoiec. Kai oi dôo eðnai perioqèc monos manthc paragontopoðhshc, ìpou ston men Z èqoume ìti kˆje akèraioc mporeð na ekfrasteð monadikˆ wc ±1 epð èna ginìmeno pr twn, ston de CI(x) kˆje polu nmo mporeð na ekfrasteð monadikˆ wc p(x) = a(x a 1 )(x a 2 )... (x a n ), me touc a, a 1, a 2,..., a n na eðnai migadikoð arijmoð. Ta parapˆnw mac dðnoun to pr to stoiqeðo thc analogðac pou diereônhse o Hensel: Oi pr toi arijmoð p Z eðnai anˆlogoi twn grammik n poluwnômwn (x a) CI(x). 8

11 H analogða pˆei akìma parapèra an skefteð kaneðc ìti dojèntoc enìc poluwnômou P (x) kai enìc sugkekrimènou a CI mporoôme, kˆnontac to anˆptugma Taylor, na grˆyoume to polu numo sth morf : P (x) = a 0 + a 1 (x a) + a 2 (x a) a n (x a) n n = a i (x a) i, a i CI. i=0 Profan c kai gia touc akeraðouc (toulˆqiston gia touc jetikoôc akeraðouc gia arq ) èqoume ìti dojèntoc jetikoô akeraðou m kai pr tou arijmoô p, mporoôme na grˆyoume ton m se bˆsh p, dhl. sth morf : m = a 0 + a 1 p + a 2 p a n p n = n a i p i, i=0 a i Z, 0 a i p 1. Autèc oi ekfrˆseic eðnai endiafèrousec giatð mac dðnoun topikèc plhroforðec: gia parˆdeigma to anˆptugma se dunˆmeic tou (x a) mac deðqnei an to a eðnai rðza tou P (x) kai ti bajmoô. Parìmoia to anˆptugma se bˆsh p deðqnoun an o m diaireðtai apì ton p kai me poiˆ pollaplìthta. T ra, gia ta polu numa kai ta phlðka touc mporoôme na poôme perissìtera prˆgmata. An pˆroume kˆpoio polu numo f(x) sto CI(x) kai èna a CI, upˆrqei pˆnta anˆptugma thc morf c: f(x) = p(x) q(x) = a n 0 (x a) n 0 + a n0 +1(x a) n = i n 0 a i (x a) i, to gnwstì apì th migadik anˆlush anˆptugma Laurent tou f(x), me touc suntelestèc a i CI kai touc ekjètec n i Z. 'Omwc autì eðnai èna pio polôploko antikeðmeno apì to anˆptugma Taylor pou qrhsimopoi same prin: To n 0 mporeð kˆllista na eðnai kai arnhtikì, gegonìc pou mac deðqnei ìti to a eðnai rðza tou q(x) kai ìqi tou p(x), diaforetikˆ, an to klˆsma den eðnai se anhgmènh morf, ìti h pollaplìthta tou a wc rðzac tou q(x) eðnai megalôterh apì aut n wc rðzac tou p(x). Me ìrouc thc Anˆlushc ja lègame ìti h f(x) èqei pìlo tˆxhc n 0 sto a. 9

12 To anˆptugma sun jwc den eðnai peperasmèno. Mˆlista ja eðnai peperasmèno tìte kai mìno tìte an to klˆsma eðnai se anhgmènh morf kai to q(x) tuqaðnei na eðnai kˆpoia dônamh tou (x a). Dhlad, sun jwc ja eðnai èna ˆpeiro ˆjroisma, kai mporeð na apodeiqjeð ìti h seirˆ f(λ) ja sugklðnei ìpote to λ eðnai arketˆ kontˆ allˆ ìqi Ðso me to a. To shmantikì eðnai ìti kˆje rht sunˆrthsh mporeð na ekfrasteð mèsw enìc anaptôgmatoc tètoiac morf c gia kˆje ènan apì touc pr touc (x a). Apì thn ˆllh meriˆ den antistoiqeð kˆje tètoia seirˆ se rht sunˆrthsh. Gia parˆdeigma oi seirèc gia to sin(x), exp(x) den apoteloôn anaptôgmata rht n sunart sewn. Dhlad èqoume dôo s mata, to CI(x) kai to s ma ìlwn twn seir n Laurent CI(x a), me to pr to na emperièqetai gn sia sto deôtero. H sunˆrthsh: f(x) anˆptugma tou f(x) gôrw apì to (x a) orðzei ton akìloujo egkleismì twn swmˆtwn CI(x) CI(x a). Upˆrqoun bèbaia ˆpeirec tètoiec apeikonðseic, mða gia kˆje a, kai kˆje mða mac dðnei topikèc plhroforðec gia th sumperiforˆ twn rht n sunart sewn kontˆ sto a. O Hensel skèfthke na epekteðnei thn analogða metaxô CI[x] kai Z ste na sumperilˆbei thn kataskeu tètoiwn anaptugmˆtwn kai sto QI. An jumhjoôme ìti to anˆlogo tou na epilèxoume èna a eðnai na epilèxoume ènan pr to p, kai ìti gia touc jetikoôc akeraðouc to zhtoômeno anˆptugma eðnai h èkfras tou se bˆsh p, mènei mìno na perˆsoume stouc jetikoôc rhtoôc. To pèrasma autì gðnetai me fusikì trìpo, grˆfontac arijmht kai paronomast se bˆsh p kai metˆ diair ntac formalistikˆ. To mìno prˆgma sto opoðo prèpei na eðmaste prosektikoð eðnai h metaforˆ, me thn ènnoia ìti dôo suntelestèc a ni, a mi gia ton ìro p i mporoôn na ajroðzoun se kˆti megalôtero tou p, opìte kai metafèroun kˆpoio kratoômeno ston epìmeno ìro p i+1. Dhlad prèpei na skeftìmaste modulo p gia touc suntelestèc kai na mhn xeqnˆme ta kratoômena. DÐnoume èna parˆdeigma tètoiac diaðreshc. Ja broôme thn 3-adik èkfrash tou rhtoô r = 32 7 : 10

13 r = 32 7 = EkteloÔme th diaðresh kai prosèqoume ìti 1 2 (mod 3): ( ) = = ( ) = = H skèyh gia thn parapˆnw diaðresh eðnai anˆlogh aut c thc diaðreshc poluwnômwn. Oi ìroi tou phlðkou a n sumplhr nontai wc ex c: prèpei na epalhjèuoun thn exðswsh: ìpou b n 1 a n b n (mod 3), oi suntèlestèc pou prokôptoun katˆ th diadikasða thc diaðreshc. Sthn parapˆnw diaðresh, ìpou b 0 = 2, b 1 = 0, b 2 = 1, b 3 = 1 k.l.p. èqoume ìti gia ta a 0 = 2, a 1 = 0, a 2 = 2, a 3 = 2 k.l.p. epalhjeôontai oi akìloujec exis seic: 1 2 = 2 (mod 3) (mod 3) 1 2 = 1 (mod 3) 1 2 = 1 (mod 3) k.o.k SuneqÐzoume th didikasða ep ˆpeiron 'Etsi èqoume thn èkfrash gia ton r: r = Deqìmenoi ìlh aut th diadikasða formalistikˆ, mporoôme na doôme ìti eðnai efarmìsimh se kˆje jetikì rhtì x = a/b kai ìti h telik seirˆ mac 11

14 deðqnei tic idiìthtec tou x se sqèsh me ton pr to p. 'Etsi, gia kˆje pr to p mporoôme na grˆyoume kˆje (jetikì mèqri t ra) rhtì arijmì a/b sth morf : x = a b = n n 0 a n p n, n 0 Z. An jewr soume ìti to klˆsma a/b eðnai se anhgmènh morf, parathroôme ta akìlouja: n 0 0 an kai mìno an p b, n 0 > 0 an kai mìno an p b kai p a n 0 < 0 an kai mìno an p b kai p a Dhlad to n 0 eðnai kˆti parìmoio me thn pollaplìthta mðac rðzac pìlou stic rhtèc sunart seic, kai se autì antanaklˆtai h pollaplìthta tou p sto a/b. QarakthrÐzetai de apì th sqèsh: x = p n 0 a 1 b 1 µɛ p a 1 b 1. Mènei na doôme p c mporoôme na pˆroume touc arnhtikoôc rhtoôc arijmoôc. Allˆ, an skeftoôme ìti oi seirèc mac mporoôn na pollaplasiastoôn, arkeð na broôme èna anˆptugma gia to 1. BrÐskoume ìti gia kˆje p isqôei: 1 = (p 1) + (p 1)p + (p 1)p 2 +, afoô eˆn prosjèsoume 1 paðrnoume: 1 + (p 1) +(p 1)p + (p 1)p }{{} 2 + = p = p + (p 1)p }{{} p 2 +(p 1)p 2 + (p 1)p 3 + = = p 2 + (p 1)p 2 }{{} p 3 +(p 1)p 3 + = = = 0 12

15 To sumpèrasma eðnai ìti mporoôme toulˆqiston tupikˆ, kaj c akìma den èqoume idèa an oi seirèc mac sugklðnoun na antistoiqðsoume se kˆje rhtì mða peperasmènh apì ta aristerˆ seirˆ Laurent dunˆmewn tou p: x = a n0 p n 0 + a n0 +1p n , n 0 Z Aut h seirˆ kaleðtai p-adikì anˆptugma tou x. Den eðnai dôskolo na deðxei kaneðc ìti to sônolo ìlwn twn apì aristerˆ peperasmènwn seir n Laurent dunˆmewn tou p eðnai s ma, ìpwc kai to CI(x a), pou eðnai s ma. JewroÔme tic akìloujec prˆxeic thc prìsjeshc kai tou pollaplasiasmoô sto sônolo twn p-adik n anaptugmˆtwn rht n arijm n: Gia x = i n 0 a i p i kai y = i m 0 b i p i orðzoume: x + y = (a i + b i )p i i min{n 0,m 0 } xy = c i p i i 2min{n 0,m 0 } me touc suntelestèc c i na eðnai c i = a i1 b i2. i 1 +i 2 =i QwrÐc blˆbh thc genikìthtac, mporoôme na gemðsoume kˆpoio apì ta dôo anaptôgmata (autì me to megalôtero apì ta n 0, m 0 ) me ìrouc me mhdenikoôc suntelestèc. PetuqaÐnoume ètsi kai oi dôo seirèc na xekinˆne apì to min{n 0, m 0 }. Bèbaia den prèpei na xeqnˆme th metaforˆ kratoumènwn, ìpou eðnai aparaðthto. Me tic prˆxeic autèc to sônolì mac gðnetai s ma, to opoðo ja sumbolðzoume me QI p kai ja to kaloôme s ma twn p-adik n arijm n. Autìc eðnai ènac trìpoc na pˆroume to QI p, kai eðnai autìc pou od ghse sthn anˆptuxh thc jewrðac twn p-adik n arijm n. Ja parousiˆsoume argìtera dôo akìma kataskeuèc, oi opoðec jmeli noun th diaisjhtik jewrða pou anaptôxame wc t ra. 'Opwc kai prin me ta s mata CI(x) kai CI(x a), h sunˆrthsh: x p-adikì anˆptugma tou x 13

16 orðzei ènan egkleismì metaxô twn dôo swmˆtwn: QI QI p. To gegonìc ìti to QI p eðnai gn sia megalôtero tou QI ja to deðxoume sthn amèswc epìmenh enìthta. SunoyÐzoume thn analogða pou melèthse o Hesnel ston akìloujo pðnaka: Z QI QI p CI[x] CI(x) CI(x a) pr toi p Z polu numa (x a) CI[x] a Z a = ±1 p 1 p 2 p n p(x) CI[x] p(x) = a(x a 1 ) (x a m ) p i pr toi Z me a, a i CI r QI r = a/b f(x) CI(x) f(x) = p(x)/q(x) me a, b Z, b 0 me p(x), q(x) CI[x], q(x) 0 dojèntoc pr tou p Z kai m Z m = a 0 + a 1 p a n p n, a i Z, 0 a i p 1 dojèntoc a CI kai p(x) CI[x] p(x) = a 0 + a 1 (x a) a m (x a) m a j CI dojèntoc pr tou p Z kai q QI dojèntoc a CI kai f(x) CI(x) q = a n0 p n 0 + a n0 +1p n , f(x) = a m0 (x a) m 0 + a m0 +1(x a) m , a i Z, 0 a i p 1, n 0 Z a j CI, m 0 Z 1.2 Exis seic modulo p n Oi p-adikoð arijmoð eðnai stenˆ sundedemènoi me to prìblhma thc epðlushc exis sewn modulo dunˆmeic tou p. Ja doôme merikˆ endiafèronta paradeðgmata pˆnw se autì to jèma. 1. MÐa exðswsh me lôseic sto QI Ja yˆxoume na broôme tic lôseic thc exðswshc x 2 25 (mod p n ) (1.1) 14

17 gia kˆje n IN. GnwrÐzoume ìti h exðswsh èqei tic akèraiec lôseic ±5. Apì autèc autìmata paðrnoume mða lôsh gia kˆje n: aplˆ paðrnoume thn x ±5 (mod p n ) gia kˆje n. Kat arq n shmei noume ìti prˆgmati gia p 2, 5 oi mìnec dunatèc lôseic wc proc isodunamða thc exðswshc x 2 25 (mod p n ) eðnai oi ±5. Autì prokôptei apì th JewrÐa Arijm n melet ntac thn perðptwsh n = 1 kai diexˆgontac ta sumperˆsmatˆ mac kai gia megalôtera n: H tetragwnik exðswsh isotimðac thc morf c x 2 a (mod p), a Z me p a, p 2 èqei eðte dôo eðte kammða lôseic. Sthn prokeimènh perðptwsh o 25 eðnai tèleio tetrˆgwno, kai ˆra èqei akrib c dôo lôseic, tic ±5. Sth sunèqeia, me epagwg sto n, apodeiknôetai ìti gia thn x 2 a (mod p n ) me p perittì pr to pou de diaireð to a, h exðswsh epidèqetai eðte dôo eðte kammða lôseic, anˆloga me th sumperiforˆ thc modulo p. Stic eidikèc peript seic ìpou eðte p a (p = 5 gia to dikì mac parˆdeigma) p = 2 èqoume ìti h exðswsh èqei pˆnw apì dôo rðzec, to pl joc twn opoðwn exartˆtai apì to n. Gia n = 1, h exðswsh x 2 a (mod p) ìtan p = 2 èqei mða lôsh, th monˆda. 'Otan p a, dhlad p = 5, kˆje x = p b, b Z eðnai lôsh. Gia uyhlìterouc ekjètec ìtan p = 2 oi lôseic thc x 2 25 (mod 2 n ) gia n 3 eðnai oi akìloujec tèsseric: ±5 kai ±5 + 2 n 1. Tèloc, mporoôme na broôme pollèc lôseic thc (1.1) kai ìtan p = 5. 'Ena aplì parˆdeigma eðnai h x 2 25 (mod 5 3 ), pou èqei, metaxô ˆllwn, lôseic tic x = ±5, ±20, ±30, ±45, ±55, ±70. Ac epistrèyoume t ra sto arqikì prìblhma kai ac prospaj soume na katano soume lðgo parapˆnw tic lôseic thc exðswshc (1.1), kai na tic susqetðsoume me thn p-adik jewrða. Ac dialèxoume èna sugkekrimèno pr to, èstw p = 3. 'Opwc eðpame h exðswsh èqei dôo lôseic. Ja broôme gia kajemða ton isoôpìloipo akèraio modulo 3 n pou an kei sto sônolo {0, 1,..., 3 n 1}. Dhlad : x 5 2 (mod 3) x (mod 3 2 ) x (mod 3 3 ) k.o.k 15

18 Autì paramènei to Ðdio ìso auxˆnei to n. MporoÔme na doôme to apotèlesma wc èna 3-adikì anˆptugma: x = Me anˆlogo trìpo, xekin ntac apì th deôterh lôsh paðrnoume kˆti pio endiafèron: x 5 1 (mod 3) x (mod 3 2 ) x (mod 3 3 ) x (mod 3 4 ) k.o.k ìpou kai pˆli mporoôme na doôme to apotèlesma wc èna ˆpeiro 3-adikì anˆptugma: x = 5 = ParathroÔme pwc h p-adik èkfrash pou prokôptei apì thn epðlush exis sewn isotimðac modulo p n gia kˆje n sqetðzontai metaxô touc. Sugkekrimèna, an kìyoume ta parapˆnw anaptôgmata ston ìro 3 i ja pˆroume mða lôsh thc exðswshc x 2 25 (mod 3 i+1 ). Mˆlista, ta anaptôgmata pou br kame me aut th diadikasða den eðnai parˆ oi 3-adikèc lôseic thc exðswshc x 2 = 25. Autì epalhjeôetai eôkola ektel ntac tic prˆxeic ìpwc tic orðsame prohgoumènwc. 2. MÐa exðswsh pou den èqei lôseic sto QI Ta prˆgmata gðnontai polô pio endiafèronta an h exðsws mac den èqei rðzec stouc rhtoôc, ìpwc gia parˆdeigma h exðswsh: x 2 2 (mod 7 n ). Gia n = 1 oi rðzec eðnai oi x 3 (mod 7) kai x 4 3 (mod 7). Gia n = 2 prèpei na skeftoôme ìti parmènec modulo 7 ja prèpei na eðnai rðzec kai gia n = 1. Dhlad, an jèsoume x = 3 + 7k kai lôsoume wc proc k èqoume: 16

19 (3 + 7k) 2 2 (mod 7 2 ) k 2 (mod 7 2 ) k 0 (mod 7 2 ) 1 + 6k 0 (mod 7) k 1 (mod 7) kai antikajist ntac mac dðnei th mða lôsh, x 10 (mod 7 2 ). OmoÐwc, th deôterh lôsh thn paðrnoume xekin ntac apì th deôterh lôsh modulo 7, thn x = 4. Aut eðnai x (mod 7 2 ). SuneqÐzontac thn Ðdia diadikasða brðskoume lôseic x 1i, x 2j pou suneqðzontai ep' ˆpeiron: x 1 = (x 11, x 12, x 13,...) = (3, 10, 108,...) kai x 2 = (x 21, x 22, x 23,...) = (4, 39, 235,...) ( 3, 10, 108,...) = x 1. 'Opwc kai sto prohgoômeno parˆdeigma, mporoôme na ekfrˆsoume ta prohgoômena wc p-adikˆ anaptôgmata: x 1 = (3, 10, 108,...) 3 = 3 10 = = kai ètsi paðrnoume thn 7-dik èkfrash: x 1 = Parìmoia brðskoume thn antðstoiqh èkfrash gia to x 2 : x 2 = Parat rhsh 1 Sta parapˆnw dedomèna axðzei na parathr soume mia leptomèreia pou ja analôsoume argìtera: gia kˆje lôsh x 1n me x 2 1 n 2 (mod 7 n ) èqoume ìti x 1n x 1n 1 (mod 7 n ). Dhlad : 17

20 10 3 (mod 7) (mod 7 2 ) (mod 7 3 ) k.o.k Parat rhsh 2 Sto s ma QI 7 h exðswsh x 2 = 2 èqei lôseic, tic x 1, x 2. SumperaÐnoume ètsi ìti to QI 7 eðnai gn sia megalôtero apì to QI. Me ta parapˆnw paradeðgmata prospajoôme na d soume èmfash sto gegonìc ìti to na lônei kaneðc exis seic modulo ìlo kai uyhlìterec dunˆmeic enìc pr tou p eðnai polô kontˆ sto na lônei thn anˆlogh exðswsh sto QI p. EÐnai mˆlista apì touc pio shmantikoôc lìgouc gia th qr sh p-adik n mejìdwn sth JewrÐa Arijm n. Melèth thc exðswshc x = 1 + 3x H parapˆnw exðswsh epilôetai eôkola kai brðskoume ìti h lôsh thc eðnai x = 1/2. An ìmwc th doôme wc èna prìblhma stajeroô shmeðou, dhlad wc èna prìblhma eôreshc lôshc miac exðswshc f(x) = x gia th sunˆrthsh f(x) = 1 + 3x, ja doôme kˆpoia polô endiafèronta prˆgmata. Tètoia probl mata epilôontai tic perissìterec forèc me kˆpoia epanalhptik mèjodo: xekinˆme apì mða arqik prosèggish x 0 kai upologðzoume thn tim thc f(x) sto x 0. Aut thn tim qrhsimopoioôme wc deôterh prosèggish x 1 = f(x 0 ) kai upologðzoume thn f(x 1 ). SuneqÐzoume me autìn ton trìpo kai parˆgoume mia akoloujða (x i ) i, h opoða elpðzoume ìti ja sugklðnei sth lôsh thc arqik c mac exðswshc. An efarmìsoume aut th diadikasða sth dik mac perðptwsh, xekin ntac apì thn arqik prosèggish x 0 = 1, ja pˆroume ta parakˆtw: x 0 = 1 x 1 = 1 + 3x 0 = x 2 = 1 + 3x 1 = x n = n 18

21 Stouc pragmatikoôc arijmoôc me th sun jh nìrma aut eðnai mia apoklðnousa akoloujða. EÐnai mða gewmetrik prìodoc me lìgo 3, thc opoðac oi ìroi gðnontai ìlo kai megalôteroi. An mporoôsame na amel soume to gegonìc ìti o lìgoc thc proìdou eðnai 3, dhlad to prìblhma ìti 3 > 1, kai qrhsimopoioôsame ton tôpo gia ˆpeiro ˆjroisma gewmetrik c proìdou, tìte ja paðrname: pou eðnai kai h swst lôsh = = 1 2, An kai sto IR h akoloujða pou sqhmatðsame eðnai apoklðnousa, den upˆrqei kˆti pou na mac empodðzei na th doôme wc akoloujða sto QI 3, me touc ìrouc thc na an koun sto QI. Profan c sto QI 3 h akoloujða (x n ) sugklðnei ston 3-adikì arijmì Autì den eðnai parˆ to 3-adikì anˆptugma tou 1/2. To endiafèron sthn perðptwsh aut eðnai ìti kˆti to apagoreumèno sto IR faðnetai na leitourgeð kalˆ ìtan to doôme apì p-adik ˆpoyh. Eisˆgontac ta s mata twn p-adik n arijm n dieurônoume thn prooptik mac kai plèon mac epitrèpontai epiqeir mata pou prohgoumènwc tan adônato na epikalestoôme. 1.3 Diofantikèc Exis seic Sto deôtero Diejnèc Sunèdrio twn Majhmatik n to 1900, o Hilbert parousðase mða lðsta apì 23 ˆluta wc tìte probl mata, diaforetikoô qarakt ra to kajèna. To mìnadikì prìblhma apìfashc apì autˆ tan to dèkato sth lðsta, kai aforoôse tic Diofantikèc exis seic. MÐa Diofantik exðswsh eðnai mða poluwnumik exðswsh f(x 1, x 2,..., x n ) = 0 me rhtoôc akèraiouc suntelestèc, sthn opoða oi aprosdiìristec mporoôn na pˆroun rhtèc akèraiec timèc. To dèkato prìblhma tou Hilbert diatup netai wc ex c: DojeÐshc mða Diofantik c exðswshc, upˆrqei peperasmènh diadikasða h opoða na apofaðnetai gia to an h exðswsh èqei lôseic? Dhlad, upˆrqei apodotikìc algìrijmoc pou na apofasðzei an mða Diofantik exðswsh eðnai epilôsimh? 19

22 To er thma apant jhke to 1970, opìte kai apodeðqjhke ìti den mporeð na upˆrxei tètoioc algìrijmoc. H apìdeixh basðsthke sth Majhmatik Logik kai th JewrÐa thc Upologisimìthtac. Pio sugkekrimèna, orðzoume wc Diofantikì SÔnolo sqetikì me kˆpoia Diofantik exðswsh f, to sônolo: {(x 1,..., x n ) IN n y 1,..., y m IN [f(x 1,..., x n, y 1,..., y m ) = 0]}. Akìma, èna anadromikˆ arijm simo sônolo A (recursively enumerable set) eðnai ekeðno gia to opoðo upˆrqei algìrijmoc pou, gia tuqaðo stoiqeðo wc eðsodo, stamatˆ se katˆstash apodoq c an kai mìno an to stoiqeðo autì an kei sto A. Tèloc, èna upologistì sônolo B (computable, recursive set) eðnai ekeðno to sônolo gia to opoðo upˆrqei algìrijmoc pou termatðzei èpeita apì peperamèno qrìno kai apofaðnetai eˆn èna stoiqeðo x an kei ìqi sto B. H apˆnthsh sto dèkato prìblhma tou Hilbert eðnai ˆmesh sunèpeia tou akìloujou apotelèsmatoc twn Yuri Matiyasevich, Julia Robinson, Martin Davis kai Hilary Putnam, gnwstì kai wc Je rhma MRDP: Kˆje anadromikˆ arijm simo sônolo eðnai Diofantikì. Epeid upˆrqei anadromikˆ arijm simo sônolo, to opoðo ìmwc den eðnai upologistì to sumpèrasma èpetai ˆmesa. Gia perissìterec leptomèreiec parapèmpoume sta [6], [2] kai [19]. 20

23 Kefˆlaio 2 Mh arqim deiec nìrmec pˆnw se s ma IK Se aut thn enìthta ja jemeli soume th jewrða pou diaisjhtikˆ anaptôxame sthn Eisagwg. Ja melet soume tic mh arqim deiec nìrmec pˆnw se èna tuqaðo s ma IK kai telikˆ ja eisˆgoume mða tètoia nìrma sto s ma twn rht n arijm n QI. Oi mh arqim deiec nìrmec èqoun polô diaforetikèc idiìthtec, se sqèsh toulˆqiston me autì pou èqoume sunhj sei wc t ra. Eisˆgoun mia diaforetik ènnoia thc apìstashc kai tou megèjouc twn pragmˆtwn me sunèpeiec mða teleðwc diaforetik topologða kai anˆlush. Allˆ ac doôme analutikˆ aut th diaforetik prosèggish. 2.1 Nìrmec pˆnw se s ma IK Orismìc 1 'Estw IK s ma. MÐa apeikìnish : IK IR + lègetai nìrma an ikanopoieð tic akìloujec idiìthtec: (i) x = 0 x = 0 (ii) xy = x y gia kˆje x, y IK (iii) x + y x + y gia kˆje x, y IK 21

24 MÐa nìrma pˆnw sto IK lègetai mh arqim deia an epiplèon ikanopoieð th sunj kh: (iv) x + y max { x, y } gia kˆje x, y IK. Diaforetikˆ ja lègetai arqim deia. Kˆje mh arqim deia nìrma sundèetai me mða ektðmhsh kai antistrìfwc. Orismìc 2 MÐa ektðmhsh (valuation) pˆnw se s ma IK eðnai mia sunˆrthsh υ : IK IR, tètoia ste gia kˆje x, y IK na isqôoun: (i) υ(0) = +, ex' orismoô (ii) υ (xy) = υ(x) + υ(y) (iii) υ(x + y) min {υ(x), υ(y)}. SugkrÐnontac tic idiìthtec (ii) kai (iii) tou orismoô thc ektðmhshc me tic idiìthtec (ii) (iv) tou orismoô thc mh arqim deiac nìrmac blèpoume ìti eðnai arketˆ parìmoiec. Oi diaforèc touc eðnai ìti to ginìmeno thc pr thc eðnai ˆjroisma sth deôterh kai ìti èqoun antðstrofh forˆ anisìthtac. MporoÔme na antistrèyoume th forˆ thc anisìthtac allˆzontac to prìshmo, kai na metatrèyoume to ˆjroisma se ginìmeno kˆnontˆc to ekjetikì. 'Etsi pernˆme apì mða ektðmhsh se mða mh arqim deia nìrma kai antistrìfwc. Genikìtera, ektðmhsh kai mh arqim deia nìrma mporoôn isodônama na apotelèsoun thn afethrða mac gia th jewrða pou ja anaptôxoume. Ja protim soume thn ènnoia thc nìrmac, an kai eidikˆ gia thn p-adik jewrða h p-adikh ektðmhsh eðnai idiaðtera shmantik. Ja parajèsoume akìma merikèc apì tic pio shmantikèc idiìthtec mðac nìrmac, arqim deiac ìqi. L mma 1 Gia kˆje nìrma pˆnw se s ma IK èqoume: (i) 1 = 1 22

25 (ii) An x IK kai x n = 1 gia kˆpoio n Z, tìte x = 1. (iii) 1 = 1 (iv) Gia kˆje x IK, x = x (v) An to IK eðnai peperasmèno s ma, tìte h eðnai h tetrimmènh nìrma. Apìdeixh: (i) An skeftoôme ìti to pedðo tim n thc nìrmac eðnai oi jetikoð pragmatikoð arijmoð kai ìti h monˆda eðnai to monadikì stoiqeðo gia to opoðo 1 2 = 1 tìte èqoume: 1 = 1 2 = = 1. (ii) ApodeiknÔetai apì th deôterh idiìthta thc nìrmac kai apì to (i). (iii) ArkeÐ na skeftoôme ìti 1 = 1 = ( 1)( 1) = 1 1, kai me to Ðdio skeptikì ìpwc sto (i) èqoume to zhtoômeno. (iv) Profanèc, ef ìson me qr sh tou (iii) èqoume: x = ( 1) x = 1 x = x. (v) ArkeÐ na jumhjoôme ìti se èna peperasmèno s ma kˆje antistrèyimo stoiqeðo k i IK mazð me thn prˆxh tou pollaplasiasmoô parˆgei kuklik upoomˆda tˆxhc n IK. Opìte, se sunduasmì me to (i) paðrnoume to zhtoômeno: 1 = 1 = ki n = k i n k i = 1 gia kˆje stoiqeðo k i IK. 23

26 Mˆlista, gia tic mh arqim deiec nìrmec isqôei kˆti polô endiafèron, pou sqetðzetai me thn idiìthta (iv) tou orismoô gia tic mh arqim deiec nìrmec: Prìtash 1 'Estw IK s ma kai mia mh arqim deia nìrma sto IK. An x, y IK me x y, tìte x + y = max { x, y }. Apìdeixh: QwrÐc blˆbh thc genikìthtac mporoôme na upojèsoume ìti x > y. Tìte, apì thn idiìthta (iv) gia mh arqim deiec nìrmec èqoume ìti: Akìma epeid x = (x + y) y, paðrnoume: x + y x = max { x, y }. (2.1) x = (x + y) y max { x + y, y }. 'Omwc, èqoume upojèsei ìti x > y, opìte h parapˆnw anisìthta mporeð na isqôei mìno an max { x + y, y } = x + y. Dhlad paðrnoume ìti x x + y kai se sunduasmì me thn (2.1) mporoôme na sumperˆnoume ìti: x = x + y. Ja parousiˆsoume t ra apì mða ˆllh optik gwnða to diaqwrismì arqim deiac kai mh arqim deiac nìrmac. Jumìmaste ìti gia kˆje s ma IK upˆrqei mia antistoðqish h : Z IK pou orðzetai wc: } {{ + 1 }, an n > 0 n n 0, an n = 0 ( ), an n < 0. }{{} n 24

27 Gia parˆdeigma, an èqoume QI IK, tìte h h den eðnai parˆ h sun jhc apeikìnish tou Z sto QI. An to IK eðnai peperasmèno, tìte h eikìna thc h eðnai èna upìswma tou IK, me plhjikìthta ènan pr to arijmì. Je rhma 1 'Estw h( Z) IK h eikìna tou Z se s ma IK. MÐa nìrma sto IK eðnai mh arqim deia an kai mìno an a 1 gia kˆje a h( Z). Sugkekrimèna, mia nìrma sto QI eðnai mh arqim deia an kai mìno an n 1 gia kˆje n Z. Apìdeixh: To eujô apodeiknôetai eôkola me epagwg : ±1 = 1 kai apì thn idiìthta (iv) gia mh arqim deiec nìrmec èqoume ìti: gia k Z k ± 1 max { k, 1}. 'Ara, an h upìjesh isqôei gia k, dhlad k 1, tìte isqôei kai gia k + 1 kai to zhtoômeno apedeðqjh. Gia to antðstrofo ja qrhsimopoi soume to anˆptugma tou NeÔtwna. Jèloume na deðxoume ìti an a 1 gia kˆje stoiqeðo tou h( Z), tìte gia opoiad - pote dôo stoiqeða tou x, y IK ja èqoume x + y max { x, y }. An kˆpoio apì ta dôo stoiqeða eðnai mhdèn tìte h anisìthta prokôptei amèswc. An ìqi, tìte mporoôme na diairèsoume me y, opìte paðrnoume thn isodônamh: { } x y + 1 max x y, 1. Dhlad, arkeð na deðxoume thn idiìthta (iv) tou orismoô thc mh arqim deiac nìrmac, me to èna apì ta dôo stoiqeða na eðnai h monˆda. 'Estw loipìn ìti kai ta dôo stoiqeða eðnai mh mhdenikˆ, y = 1 kai èstw m opoiosd pote jetikìc akèraioc. Tìte èqoume: x + 1 m = ( m ) k k x k ( ) m x k k k k x k (m + 1) max { x m, 1} (apì upìjesh, afoô ( m n) Z ) 25

28 PaÐrnontac thn m-ost rðza kai sta dôo mèlh paðrnoume: x + 1 m m + 1 max {1, x }, anisìthta pou isqôei gia kˆje jetikì akèraio m. 'Omwc gnwrðzoume ìti lim m m m + 1 = 1, opìte an af soume to m na trèxei sto ˆpeiro èqoume to zhtoômeno: x + 1 max { x, 1}. Ekfrˆzontac to parapˆnw je rhma lðgo diaforetikˆ, autì pou mac lèei eðnai to akìloujo: Pìrisma 1 MÐa nìrma eðnai mh arqim deia an kai mìno an sup { n : n Z} = 1. Gia na oloklhr soume th sôgkrish metaxô miac arqim deiac kai miac mh arqim deiac nìrmac ja qarakthrðsoume tic arqim deiec nìrmec wc autèc pou èqoun thn akìloujh idiìthta: Arqim deia idiìthta: Dojèntwn x, y IK, x 0, upˆrqei jetikìc akèraioc n tètoioc ste nx > y. Ousiastikˆ, h parapˆnw idiìthta mac lèei ìti metr ntac to mègejoc twn akeraðwn me mia arqim deia nìrma, mporoôme na pˆroume aujaðreta megˆlouc akèraiouc. Me ˆlla lìgia an h eðnai arqim deia tìte: sup { n : n Z} = +. Tèloc, axðzei na anafèroume ìti autèc oi dôo peript seic eðnai kai oi monadikèc dunatèc. Prˆgmati, lìgw thc pollaplasiastik c idiìthtac thc nìrmac, eˆn upˆrqei kˆpoioc akèraioc k me nìrma megalôterh tou èna, tìte lim n kn = lim k n = +, n kai ˆra: sup { n : n Z} = +. An den upˆrqei tètoio k tìte, ef ìson 1 = 1, ja èqoume sup { n : n Z} = 1. 26

29 2.2 'Ena parˆdeigma: h p-adik nìrma sto QI Ja xekin soume apì thn p-adik ektðmhsh sto QI gia na katal xoume sthn p- adik nìrma sto QI, wc èna parˆdeigma mðac mh arqim deiac nìrmac pˆnw se s ma. Autì ja eðnai pou ja mac apasqol sei se aut thn ergasða. Orismìc 3 'Estw ènac pr toc p Z. H p-adik ektðmhsh sto Z orðzetai wc h sunˆrthsh υ p : Z\ {0} IN {0} tètoia ste: gia kˆje mh mhdenikì n Z, h υ p (n) eðnai h mègisth dônamh tou p pou ton diaireð, dhlad o monadikìc mh arnhtikìc akèraioc pou ikanopoieð th sqèsh: n = p υp(n) n, me p n. MporoÔme na epekteðnoume thn ektðmhsh sto s ma twn rht n wc akoloôjwc: Orismìc 4 H p-adik ektðmhsh sto QI eðnai h epèktash thc υ p pou orðzetai wc ex c: υ p : QI \ {0} IR, tètoia ste an x = a/b QI \ {0} me a, b Z, b 0, h p-adik ektðmhsh tou x dðnetai apì th sqèsh: υ p (x) = υ p (a) υ p (b). Gia to 0 orðzoume υ p (0) = +. Se antistoiqða me ton Orismì 3 mporoôme na doôme ìti gia thn p-adik ektðmhsh enìc rhtoô arijmoô x isqôei: x = p υp(x) a b, p ab. Je rhma 2 H p-adik ektðmhsh eðnai ektðmhsh, ìpwc aut orðsthke ston Orismì 2. 27

30 Apìdeixh: To mìno pou qreiˆzetai na kˆnoume eðnai na epalhjeôsoume tic idiìthtec thc sunˆrthshc ektðmhshc. Ex' orismoô èqoume ìti υ p (0) = +. Gia tic idiìthtec (ii) kai (iii) èqoume ìti an x = a/b, y = c/d QI tìte: υ p (xy) = υ p ( a c b d ) = υ p( ac bd ) = υ p(ac) υ p (bd) = = υ p (a) + υ p (c) υ p (b) υ p (d) = = υ p (x) + υ p (y). Epiplèon, me υ p (x) = n, υ p (y) = m èqoume: υ p (x + y) = υ p (p n a b + p m c d ) = υ p(p min{n, m} x ) min {n, m}, afoô o rhtìc x mporeð na diaireðtai ìqi apì ton p. Orismìc 5 Gia kˆje x QI orðzoume thn p-adik nìrma tou x wc: x p = p υp(x) an x 0, kai 0 p = 0. Parat rhsh 3 O orismìc gia th nìrma tou mhdenìc eðnai apolôtwc sumbatìc me ton orismì gia thn ektðmhsh sto mhdèn. Parat rhsh 4 AxÐzei na prosèxoume ìti to pedðo tim n thc p-adik c nìrmac eðnai to sônolo {p n : n Z}. Dhlad, èqoume diakritì kai arijm simo pedðo tim n. Mènei na deðxoume ìti autì pou orðsame wc p-adik nìrma eðnai prˆgmati mia nìrma. Je rhma 3 H sunˆrthsh p eðnai mða mh arqim deia nìrma sto QI. Apìdeixh: Oi idiìthtec thc nìrmac prokôptoun ˆmesa, efarmìzontac to L mma 1 gia thn p-adik ektðmhsh. 28

31 Wc parˆdeigma, ac upologðsoume ta akìlouja: υ 7 (902), υ 7 (902/35), υ 5 (400) kai tic antðstoiqec p-adikèc nìrmec. 902 = υ 7 (902) = 0 35 = 5 7 υ 7 (35) = 1 } υ 7 (902/35) = υ 7 (902) υ 7 (35) = = υ 5 (400) = 2 kai gia tic nìrmec twn parapˆnw stoiqeðwn èqoume: = 7 υ7(902) = 7 0 = 1 7 = 7 υ7(902/35) 7 ( 1) = = 5 υ 5(400) = Oi diaforetikèc nìrmec sto QI Ac doôme ti akrib c kˆnei h p-adik nìrma: ìso pio polu ènac arijmìc x diaireðtai apì ton pr to p pou èqoume epilèxei, tìso pio megˆlh eðnai h ektðmhs tou υ p (x) kai tìso mikrìterh h p-adik tou nìrma x p. H p-adik nìrma èqei ènan teleðwc diaforetikì trìpo mètrhshc tou megèjouc enìc arijmoô se sqèsh me th sun jh nìrma sto QI. H pr th mac lèei ìti ènac arijmìc eðnai mikrìc ìtan diaireðtai polô me ton p, en h deôterh ìti eðnai mikrìc ìtan eðnai kontˆ sto mhdèn. Ja kleðsoume aut thn enìthta anafèrontac ìti oi dôo autoð tôpoi norm n eðnai oi monadikèc, wc proc isodunamða, mh tetrimmènec nìrmec pou mporoôn na oristoôn sto QI. SumbolÐzoume th sun jh apìluth tim sto QI wc. EÐnai bolikì na skeftìmaste to sômbolo san ènan pr to arijmì sto Z kai na anaferìmaste se autìn wc o ˆpeiroc pr toc, kurðwc diìti mporoôme ètsi na uiojet soume to sumbolismì p gia kˆje p. Mˆlista, eðnai polô endiafèron to akìloujo: Je rhma 4 Gia kˆje x QI \ {0}, èqoume ìti: x p = 1. p 29

32 Apìdeixh: ArkeÐ na deðxoume to zhtoômeno gia ènan jetikì akèraio, kaj c ta upìloipa èpontai eôkola apì autì. GnwrÐzoume apì to Jemeli dec Je rhma thc Arijmhtik c ìti ènac jetikìc akèraioc x grˆfetai monadikˆ wc ginìmeno dunˆmewn pr twn: x = p a 1 1 p a p a k k. Tìte gia thn p-adik nìrma tou x èqoume ìti: 1, anp p i gia kˆje i x p = p a i i, an p = p i gia kˆpoio i = 1, 2,..., k x = p a 1 1 p a p a k k, an p = To ìti to ginìmeno ìlwn twn p-adik n norm n tou x èinai Ðso me th monˆda prokôptei ˆmesa. Dedomènou ìti, orðzontac mia nìrma pˆnw se èna s ma IK, tautìqrona epˆgetai kai mia topologða se autì, ja orðsoume pìte dôo nìrmec eðnai isodônamec wc akoloôjwc: Orismìc 6 DÔo nìrmec 1 kai 2 pˆnw se s ma IK eðnai isodônamec an orðzoun thn Ðdia topologða sto IK, dhlad eˆn kˆje sônolo pou eðnai anoiktì wc proc th mða nìrma eðnai anoiktì kai wc proc thn ˆllh. Pernˆme ètsi sto Je rhma Ostrowski, pou mac lèei ìti èqoume brei ìlec tic nìrmec pˆnw sto QI, to opoðo parajètoume qwrðc apìdeixh. Je rhma 5 (Ostrowski) Kˆje mh tetrimmènh nìrma sto QI eðnai isodônamh me kˆpoia apì tic nìrmec p, ìpou o p eðnai eðte ènac pr toc arijmìc p =. 2.4 TopologÐa se s ma me mh arqim deia nìrma Kˆje nìrma perikleðei mða ènnoia megèjouc twn stoiqeðwn sta opoða efarmìzetai. EpÐshc epˆgei mia metrik sto s ma mac, dhlad mporoôme na metr - soume apostˆseic metaxô twn stoiqeðwn tou. 'Eqontac th metrik mporoôme 30

33 na orðsoume anoiktˆ kai kleistˆ sônola, na melet soume th sunektikìthta tou s matoc, kai genikìtera na ereun soume thn topologða pou orðzetai se autì. 'Otan h nìrma mac èqei idiaðterec idiìthtec, tìte kai h metrik pou epˆgetai dðnei sto q ro idiaðterh topologða. Aut thn idiaðterh topologða twn mh arqim deiwn norm n ja melet soume se aut thn enìthta. Orismìc 7 'Estw IK s ma kai opoiad pote nìrma sto IK. OrÐzoume wc apìstash d(x, y) metaxô dôo stoiqeðwn x, y IK wc: d(x, y) = x y. H sunˆrthsh d(x, y) kaleðtai metrik pou epˆgetai apì th nìrma. metrik ikanopoieð tic akìloujec idiìthtec: MÐa (i) gia kˆje x, y IK isqôei: d(x, y) 0 kai d(x, y) = 0 x = y (ii) gia kˆje x, y IK isqôei: d(x, y) = d(y, x) (iii) gia kˆje x, y IK isqôei: d(x, z) d(x, y) + d(x, z). H teleutaða idiìthta kaleðtai kai trigwnik anisìthta, kaj c ekfrˆzei to gegonìc ìti to ˆjroisma dôo pleur n enìc trig nou eðnai pˆnta megalôtero Ðso apì thn trðth pleurˆ. 'Enac q roc pˆnw ston opoðo eðnai orismènh mða metrik kaleðtai metrikìc q roc. H parapˆnw prìtash mac lèei ìti opoiod pote s ma pˆnw sto opoðo èqei oristeð mða nìrma metatrèpetai se metrikì q ro an orðsoume th metrik d(x, y) ìpwc parapˆnw. Mˆlista aut h metrik sumperifèretai kalˆ wc proc tic prˆxeic tou s matoc. Sugkekrimèna, h prìsjesh, o pollaplasiasmìc kai to antðstrofo stoiqeðo eðnai wc proc th d(x, y) suneqeðc sunart seic, kai to s ma lègetai topologikì s ma. Prìtash 2 H metrik d(x, y) = x y pou epˆgetai apì th nìrma sto s ma IK kˆnei to IK topologikì s ma. Apìdeixh: Ja deixoume ìti h prìsjesh, o pollaplasiasmìc kai to antðstrofo stoiqeðo eðnai suneqeðc sunart seic. 31

34 H sunˆrthsh thc prìsjeshc + : IK IK IK eðnai suneq c sto (x 0, y 0 ) an kai mìno an gia kˆje ɛ > 0 upˆrqei δ > 0, tètoio ste opoted pote d(x, x 0 ) < δ kai d(y, y 0 ) < δ tìte d(x + y, x 0 + y 0 ) < ɛ. Prˆgmati, gia kˆje ɛ > 0 mporoôme na epilèxoume δ = ɛ/2 > 0 kai èqoume: d(x + y, x 0 + y 0 ) = (x + y) (x 0 + y 0 ) = (x x 0 ) + (y y 0 ) x x 0 + y y 0 = δ + δ = ɛ ParathroÔme kˆti pou ja qrhsimopoi soume parakˆtw: gia kˆje δ ste d(x, x 0 ) = x x 0 < δ isqôei: x x 0 x x 0 < δ δ < x x 0 < δ x 0 δ < x < x 0 + δ. H sunˆrthsh tou pollaplasiasmoô : IK IK IK eðnai suneq c sto (x 0, y 0 ) an kai mìno an gia kˆje ɛ > 0 upˆrqei δ > 0 tètoio ste opoted pote d(x, x 0 ) < δ kai d(y, y 0 ) < δ tìte d(xy, x 0 y 0 ) < ɛ. Gia kˆje ɛ mpor na epilèxw δ = a+ a 2 + ɛ > 0, ìpou a = max { x 0, y 0 } kai ɛ lðgo mikrìtero tou ɛ. 'Etsi prokôptei: d(xy, x 0 y 0 ) = xy x 0 y 0 = xy xy 0 + xy 0 x 0 y 0 x y y 0 + y 0 x x 0 = x δ + y 0 δ (δ + x 0 )δ + y 0 δ = ( a + a 2 + ɛ + a)( a + a 2 + ɛ ) + y 0 a + a 2 + ɛ = a 2 + ɛ ( y 0 a) + a( y 0 a) + ɛ (a + a )( y 0 a) + a( y 0 a) + ɛ, me a < ɛ = a ( y 0 a) + ɛ ɛ < ɛ. H sunˆrthsh tou antistrìfou h : IK IK eðnai suneq c sto x 0 an kai mìno an gia kˆje ɛ > 0 upˆrqei δ > 0, tètoio ste opoted pote d(x, x 0 ) < δ tìte d(1/x, 1/x 0 ) < ɛ. Gia kˆje ɛ mporoôme na epilèxoume δ = ɛ x ɛ x 0, ìpou ɛ lðgo mikrìtero tou ɛ, kai ètsi èqoume: 32

35 d( 1, 1 x x 0 ) = 1 1 x x 0 = x 0 x x 0 x = d(x, x 0) x x 0 < δ x x 0 < = δ ( x 0 δ) x 0 ɛ x 0 2 (1 ɛ x 0 ) x 0 2 ɛ x 0 2 x 0 = ɛ < ɛ. Prìtash 3 MÐa nìrma pˆnw se s ma IK eðnai mh arqim deia an kai mìno an gia th metrik pou aut epˆgei sto s ma isqôei: d(x, z) max {d(x, y), d(y, z)} gia kˆje x, y, z IK. Apìdeixh: Gia to eujô, apl c efarmìzoume th mh arqim deia idiìthta (bl. Orismì 1 (iv) ) sthn isìthta: (x z) = (x y) + (y z). Gia to antðstrofo, epilègoume y = y 1 kai z = 0 kai ta antikajistoôme sthn anisìthta, opìte èqoume: d(x, y) max {d(x, z), d(z, y)} d(x, y 1 ) max {d(x, 0), d(0, y 1 )} dhlad : x + y 1 max { x, y 1 } gia kˆje x, y 1 IK. H anisìthta sthn prìtash eðnai gnwst wc oultrametrik anisìthta kai ènac metrikìc q roc ston opoðo aut epalhjeôetai lègetai oultrametrikìc. Prìtash 4 Se ènan oultrametrikì q ro IK ìla ta trðgwna eðnai isoskel. Apìdeixh: 'Estw x, y, z IK. MporoÔme na doôme tic metaxô touc apostˆseic wc ta m kh twn pleur n enìc trig nou: d(x, y) = x y d(y, z) = y z 33

36 d(x, z) = x z. Jumìmaste thn Prìtash 1 pou mac lèei ìti gia mia mh arqim deia nìrma isqôei: gia x, y IK an x y tìte x y = max { x, y }. Sth dik mac perðptwsh èqoume ìti: (x y) + (y z) = (x z) (x z) (y z) = (x y), gia kˆje x, y, z IK. Epomènwc, an dôo pleurèc eðnai ˆnisec, èstw d(x, z) d(y, z), tìte h trðth pleurˆ, h d(x, y), eðnai Ðsh me th megalôterh apì tic dôo. Se kˆje perðptwsh, afoô ta x, y, z mporoôn na enallˆssontai sthn parapˆnw isìthta, dôo apì tic treic pleurèc eðnai Ðsec, dhlad, kˆje trða shmeða sqhmatðzoun isoskelèc trðgwno. Mia shmantik ènnoia stouc metrikoôc q rouc eðnai aut thc mpˆlac. Se ènan oultrametrikì q ro oi mpˆlec pou orðzontai èqoun idiìthtec pou ja mac fanoôn asun jistec. 'Estw a IK kai r IR +. SumbolÐzoume me B(a, r) thn anoikt mpˆla me kèntro to a kai aktðna r, en sumbolðzoume me B(a, r) thn kleist mpˆla me kèntro to a kai aktðna r. Prìtash 5 'Estw (IK, ) topologikì s ma me mh arqim deia nìrma. Tìte isqôoun: (i) Gia kˆje b B(a, r) isqôei B(a, r) = B(b, r). Dhlad kˆje shmeðo miac anoikt c mpˆlac eðnai kèntro thc mpˆlac. (ii) Gia kˆje b B(a, r) isqôei B(a, r) = B(b, r). Dhlad kˆje shmeðo miac kleist c mpˆlac eðnai kèntro thc mpˆlac. (iii) To sônolo B(a, r) eðnai anoiktì kai kleistì sônolo. (iv) An r 0, to sônolo B(a, r) eðnai anoiktì kai kleistì sônolo. (v) 'Estw a, b IK kai r, s mh mhdenikoð jetikoð arijmoð. IsqÔei B(a, r) B(b, s) 0 an kai mìno an B(a, r) B(b, s) B(a, r) B(b, s). Dhlad kˆje dôo anoiktèc mpˆlec eðte ja eðnai xènec metaxô touc h mða ja perièqetai sthn ˆllh. 34

37 (vi) 'Estw a, b IK kai r, s mh mhdenikoð jetikoð arijmoð. IsqÔei B(a, r) B(b, s) 0 an kai mìno an B(a, r) B(b, s) B(a, r) B(b, s). Dhladh kˆje dôo kleistèc mpˆlec eðte ja eðnai xènec metaxô touc h mða ja perièqetai sthn ˆllh. Apìdeixh: (i) Ex orismoô èqoume: b B(a, r) b a < r. An t ra pˆroume opoiod pote x B(a, r), apì th mh arqim deia idiìthta paðrnoume: x b = x a + a b max { x a, b a } < r, dhlad x B(b, r) kai ˆra B(a, r) B(b, r). Gia na deðxoume ìti kai B(a, r) B(b, r), arkeð na allˆxoume ta a kai b sthn parapˆnw anˆlush. 'Etsi oi dôo mpˆlec eðnai Ðsec. (ii) Akrib c ta Ðdia b mata me to (i), mìno pou antð gia < bˆzoume. (iii) Kˆje anoikt mpˆla se metrikì q ro eðnai anoiktì sônolo. Autì pou jèloume na deðxoume eðnai ìti sth mh arqim deia perðptwsh eðnai epðshc kleistì. Prˆgmati, èstw x oriakì shmeðo thc B(a, r). Dhlad isqôei ìti gia kˆje ɛ > 0, B(x, ɛ) 0/. Akìma, èstw s r kai èstw h antðstoiqh mpˆla me kèntro x kai aktðna s, B(x, s). oriakì shmeðo, tìte: AfoÔ to x eðnai B(a, r) B(x, s) 0/ y B(a, r) B(x, s) y a < r kai y x < s r. Efarmìzontac th mh arqim deia idiìthta paðrnoume: x a max { x y, y a } < max {s, r} = r, dhlad x B(a, r). Dhlad èqoume ìti kˆje oriakì shmeðo thc B(a, r) an kei sthn B(a, r), kai ˆra h B(a, r) eðnai èna kleistì sônolo. 35

38 (iv) H apìdeixh ìti h mpˆla B(a, r) eðnai kleist eðnai akrib c ìpwc tou (iii). Ja deðxoume ìti eðnai kai anoikt. 'Estw x B(a, r) kai èstw s = x a r. Jètoume r = r s r. Tìte, B(x, r ) B(a, r). Prˆgmati, èstw y B(x, r ). Tìte èqoume: y a = y x + x a max { y x, x a } r, kai ˆra y B(a, r). Prosèqoume th leptomèreia r 0. An r = 0 tìte h kleist mpˆla me kèntro x kai aktðna 0 eðnai to monosônolo {x}, to opoðo eðnai kleistì allˆ ìqi anoiktì, afoô den upˆrqei mh ken anoikt mpˆla me kèntro to x pou na perièqetai gn sia sto {x}. (v) QwrÐc blˆbh thc genikìthtac upojètoume ìti s r. An oi dôo mpˆlec den eðnai xènec, tìte upˆrqei kˆpoio c B(a, r) B(b, s). Tìte, gnwrðzoume apì to (i), ìti B(a, r) = B(c, r) kai B(b, s) = B(c, s). Dhlad : B(b, s) = B(c, s) B(c, r) = B(a, r), kai to zhtoômeno apedeðqjh. (vi) Akrib c ìpwc to (iv) me qr sh tou (ii). Pìrisma 2 H sfaðra S(a, r) = {x IK : x a = r} eðnai anoiktì kai k- leistì sônolo. Apìdeixh: Einai anoiktì diìti, an pˆroume tuqaðo shmeðo x S(a, r) kai 0 ɛ < r, tìte h B(x, ɛ) S(a, r), afoô gia kˆje y B(x, ɛ) èqoume apì Prìtash (1): y a = y x + x a = max { y x, x a } = r, afoô y x x a. EÐnai kleistì, diìti eðnai h tom dôo kleist n sunìlwn, twn B(a, r) kai IK\B(a, r). 36

39 To gegonìc ìti upˆrqoun tìsa pollˆ anoiktˆ-kleistˆ sônola se ènan oultrametrikì q ro (kˆje mpˆla pou mporeð na oristeð eðnai anoikt -kleist ) mac lèei pollˆ prˆgmata gia thn topologða tou. KleÐnontac aut thn parˆgrafo ja melet soume th sunektikìthta se ènan tètoio q ro, qrhsimopoi ntac pollˆ apì ta apotelèsmata thc prohgoômenhc anˆlushc. Orismìc 8 'Ena sônolo S kaleðtai mh sunektikì an mporoôn na brejoôn dôo anoiktˆ sônola S 1, S 2, tètoia ste: (i) S 1 S 2 = 0/, (ii) S = (S S 1 ) (S S 2 ), (iii) ta S 1, S 2 eðnai mh kenˆ. Prìtash 6 Se s ma IK me mh arqim deia nìrma kˆje kleist mpˆla aktðnac r > 0 eðnai mh sunektik. OmoÐwc kai gia tic anoiktèc mpˆlec. Apìdeixh: Gia kˆje kleist mpˆla B(a, r) isqôei ìti mporeð na grafeð wc ènwsh dôo anoikt n, mh ken n kai xènwn metaxô touc sunìlwn wc ex c: B(a, r) = B(a, r) {x IK : x a = r}. Gia kˆje mh ken anoikt mpˆla B(a, r) mporoôme na epilèxoume s < r kai na sqhmatðsoume ta ex c anoiktˆ mh kenˆ sônola: B(a, s) kai B(a, r)\b(a, s). To pr to sônolo eðnai anoiktì epeid eðnai mpˆla. To deôtero wc sumpl - rwma enìc kleistoô sunìlou se anoiktì sônolo. MporoÔme na epilèxoume to s ste kai ta dôo sônola na eðnai mh kenˆ. Profan c ta dôo sônola eðnai xèna metaxô touc kai h ènws touc mac dðnei ìlh thn anoikt mpˆla. Sunep c kˆje anoikt mpˆla eðnai mh sunektikì sônolo. Orismìc 9 'Estw x IK. OrÐzoume wc sunektik sunist sa tou x na eðnai h ènwsh ìlwn twn sunektik n sunìlwn pou perièqoun to x. IsodÔnama eðnai to megalôtero sunektikì sônolo pou perièqei to x. 37

40 Prìtash 7 Se s ma IK me mh arqim deia nìrma h sunektik sunist sa kˆje shmeðou x IK eðnai to monosônolo {x}. Dhlad to IK eðnai olikˆ mh sunektikì topologikì s ma. Apìdeixh: 'Estw ìti h sunektik sunist sa S tou x perièqei kˆpoio y x kai èstw r = x y h apìstash twn dôo stoiqeðwn. Tìte oi dôo mpˆlec B(x, r/2)kai B(y, r/2) eðnai dôo sônola pou plhroôn tic idiìthtec tou orismoô thc mh sunektikìthtac. 'Eqoume: S {x, y} = (S B(y, r/2)) (S B(x, r/2)). Epiplèon ènai xènec metaxô touc, afoô eˆn eðqan kˆpoio koinì stoiqeðo, tìte apì thn Prìtash 5 h mða ja perieðqeto sthn ˆllh. Tìte ìmwc x y r/2, to opoðo eðnai ˆtopo, afoô x y = r > r/2. Tèloc eðnai anoiktˆ sônola, wc mpˆlec, kai mh kenèc, afoô perièqoun ta stoiqeða x kai y antðstoiqa. Sunep c kˆje sônolo megalôtero apì to monosônolo paôei na eðnai sunektikì. 2.5 TopologÐa sto QI me thn p-adik nìrma EÐdame p c ìla ta trðgwna eðnai isoskel se s ma IK me mða mh arqim deia nìrma. Ac doôme t ra p c faðnetai h idiìthta aut sto QI me thn p-adik nìrma. Ja pˆroume treic akeraðouc x, y, z kai tic metaxô touc apostˆseic: α = x y, β = y z kai γ = x z. Ja deðxoume ìti pˆnta dôo apì autèc ja eðnai Ðsec. Profan c α, β, γ Z kai isqôei γ = α + β. Gia tic nìrmec touc èqoume: α p = p υp(α) = p n, β p = p υp(β) = p m kai γ p = p υp(γ) = p k, ìpou υ p (α) = n, υ p (β) = m, υ p (γ) = k. IsodÔnama, mporoôme na grˆyoume touc α, β, γ wc: α = p n α, β = p m β, γ = p k γ, me p α β γ. 'Estw α p > β p. Tìte n < m kai èstw m = n + ɛ. Gia to γ èqoume: 38

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