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Av alb xa ble tote alb+c Ocuouóe 8

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d = đ1Z1 + - + „0

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av o pĩyiotos Kowds d1aipĩtns tous «tvai to 1

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oto d va etvar ưo UếYtGtoc XoWĩc ðt@toểtfc Tov a,p Toted=1%7d=p, exer] o p eiva Tomtoc xa ồev yết đÀÀoUC Staipétec Av d = p xatadryouue oto étt o p Stalpet tov a Avd = 1 Ba undeyouv dv0 dxépotot Z,1/ ue 1 = ax + py dpa

b = abx+ pay xa apov o p diatpet tov ab Eyouue ducoa Ott Statpet tov b Le xade

tr] ƠcIpÁ T@V gì, dạ ƠC đị, đ„ OaTE va EvoupE Pi = G, - Dn = G, °

Anoddetgy: To yeyovdec ot: xade Vetixd¢ axépaiog a > 1 yedweta 0v YIVĨUEVO TO@)t@V ŒxtOồet⁄VÚetơdt cÚXoÀd te cnrodyeoyf Tia to 2 toyver apov o idtoc etvou Tewtoc Ác UuroÙéơouUe ĩtt YV@Of€oUUe OTL ot 2, ,7 avadvovTaL Of yIvGUEVO TOOTWV KO aC EYEeT&oOOUUE Tt OUUBatver yla Tov 2+ 1 Ette autd¢ etvor me@toc

HO EYOUUE TỀEt@OEt cite Sev civar xa Guvena@c Va YoOdpETa Gav Eva YIvOUEVO N+1l=nyNng Ue 2 € n1,N2 <n O xadEvac and aUTOUS toucg aprOyouc Da civau YIVOUEVO TOPOTWY (ATO THY ETAYwYIXY Yao UTd6VEON, 21 = Pi De, Ne = Gi - Gm ) doa xmron+1l=p, ,Deqi -Am Va civar extone

[ia To LOVOOTUAVTO TS avaTapdoTAONS TAapATHPOUUE ro cŠfc Ac UnoVECOUUE OTLO N= DPi .Pn = N1 -Gm elye SbO avanapacThoElS OE YIVOUEVO TEM@TWV TOUG omotoug éyouUe yee: ue œÚÊovtd toĩro SndadH pi < po , Gi < qo “mL extong O11 nm < m LIoyupiCéuaote 611 mM =n xa pi = Gy Ya xaVe 7 = 1, 7 Tlokyuatt, av auté mov toyueiCduaote dev foyve dbo nepintwoEIc Da UTOpOvCaY

®Me tov d90 <unopovue va oAAdEouuE trị ơetođ> evvooÚue ĩtt utopo0ue vớ ƯooÚue ula Ue- xớĐcơn (ồnÀdồi, uuv 1-1 xĩt ent ameixdvion) wm: {1, ,n} 4 {1, ,m} we pr = Ga(tys -59n = Gr(n): Eva dkhog toosvvaoc teémo¢ va ulAhoouue yia Lovadixdtyta efvar va Ưe@po0ue ĩtt otyny œvVớÀUon vĩc dovUoÚ @ = G1 Pm Ol TEW@TOL eupaviGovta UE avfovoa oelod, SyAady

Pi < po < xa avutd efvan mov Va yonoworotyjoouue otyy anddergn H diatdmwon auth duwe Sev utopel va uetapepvel of yeuxotc SaxtbALouc 6Tou Sev EyouuE xdTOLA OYEoN OAK SdTAENC UETAEY TwWY OTOLyElwY TOUC.

va ouuBatvouy:

Kite pj = q yra xede 2 = 1, ,n xan < m, aAAd Tote Va XœtdĂfydUe OT

1 = Gn41 +Gm Tou eivar paveod& &tono

Kite Qa UuxớpXet xớnoto Ì < k < 0m te 0y # qe Tote Va ciyaue pe Pn = k - - - Jm-

ĐV 0y < dy tốtE TO DE < đ;¡ Yt ÓĂœŒ tớ ý = k,k + 1, ,?m AXĂĩ to ÐgÌqk - - - đm xơt ơUVex@c 0% Unớoyet xớnotO ý = k,k + 1, ,m ye pg|d Enetồf óuoc O gị elvan Tew@toc Va eiyaue OTL D_y = Gi, &tToTO Av TOAL gy < Øy TÓtE đy < Đ¿ Yt% XớÙ€ t=k, ,2 OAK gelDe - Dn HO TOAL Va Experts va Statpet xdmOLOV ANd TOUS P; UE tov oroto xa Ya tauTiICĩtayv

Katodngaue mdr of &tono xo to Ocwoenua anodetytynxe O

Aró to Teonyouuevo Vewonua Va Eyouue duce to e€y<:

Ilpóvvơn xœi Optouóec 1

(a) Kade Oetixds axĩpais n > 2 ypdpetai Katd povadixd tpoóno 0tr) Ho0@11:

— myử1 Uk

N=Ppy D,

OMOU 01 P1, -, De €tvat Tpa@tor apiOpol we Py < +++ < Py KQ1 Ol Uy, ,UK AKEpaiol

peyadutepor 4) toot tov 1 H napdotaon aut Ăĩy€t01 rịỊ avdAvon tov n oe VIVÓJL€TO TO(C0

(B) KâÖ< akâpmoe n # Ú,1 ypoâ@ctat Katâ HoV/dỗtKÓ tOÓn0 otny proper:

n = sing(a)py' .p,*

OTOU O1 Pi, ,D_ €tval TpoTtO1 apIOuot pe py < +++ < Pp, Ol U1, , UK AKEpaiol

peyadutepoi 7) tooi tou 1”

Ox TEASLMOOUUE AUTH THY Eloaywyixh Tapcypapo opiCovtac yra xaVe aordue n Ula TOAD YOON Yla Ta Tapaxdtw oyĩon tooduvauiag oto cbvoAO Z twv axepatov Tia va efvou &cxứỦœoo wo Tpo¢ Toto apiOud n Vewoovus TH oyĩon tooduvautac

yenowornoiovue tov xdnw>o aouvyioto ouuBodtoud - = (mod n) ya tH OYEON AUTH

Ocuouóec 12

Eotw n ÖctK€Óc aKÓ0đioG H€aĂÚtSpo€ 1) toog tou 2 Av a,b € Z Ủa ypdgoupe

ớt

a=b (mod n)

av Kai jidvo av o n diapet toy a — b

H oyĩon auty eivor uta oyĩon tooduvautac otouc axĩeatouc Authy xa đĂĂec Ơtoty£t@ðetc tồtótritec trịc oyĩonic d0trfc øUuvofŠouue ơtrịV EXdUEVN TEdTAON;:

Ileórvơn 2

Eotw n betixds axĩpatog peyadttepos 4 toos tov 2 cai a,b,c,a',b',c € Z Tote:

a =a (mod n)

Ava=b (mod n) tĩte b= a (mod n)

Ava=b (mod n) Kơi b = c (mod n) tớcc ø = c (mod n)

Avaz=da' (mod n) ai b = Ù'` (mod n) cóc a +b = ø +Ù (mod n)

Ava=ư (mod n) kœib =bÙ' (mod n) cóc ab = a’b! (mod n)

Tìa kâÖc a € 2 UunâpX«t Hovaôicóc j € {0,1, ,n — 1} we 7 =a (mod n)

Arnddetĩy: Ac Seifouue to (5) Av Av a = a! (mod n) xa b = Ỉ (mod n)

tótg Yt% xớnotd k,l € Z Va ĩyouue 611 a = kn +a’,b = In+', doa ab =

(kln + kb! + la')n + ad’, 8ndadh ab = a'b' (mod n) To (6) civor ducon ouvenera

THS TaUTOTHTAS TH Statpconc (Oewenua 4)

H anddetcn tav unodoinwy agrveta otov avayvootyn OF

Aoxnon 4 Na anodeifete tov povadixdtnta tov k,v oto Ocwpnyua 44

Aoxyor 5 Na Bpeite to peyioto Kow6d diaypĩtn (a,b) tay a,b Kai va tov exgpd-

O€TE OTN Popp xa + yb ay

Aoxnon 7 Na detéete 6t1 Evas puoikds apiOuds n €tvai mpa@tos av Kai povo ay dev diapettar and Kavévay np@to p < vn

Aơxroơn 8 Na delete 61 av m,n etvai un pndevixot axépaioi tote (m,n) = (|m|, |n|)

Aoxyon 9 Na detéete 61 av (m,n) = 1 xark € Z\{0} téte (km, kn) = |k|(m, n)

Aoxnorn 10 Eotw n tetixds axépaios Aciéte 6t1 av évas tpatos p diaipet tov

ni+1 tote p> n

Aơxrnơn 11 Na dettete 61 urdpyovuy drepoi mpaetoi apOuot ‘vas todos etvai

Va KPNOWOTOINOETE THY TpoNnyovpevNn doxnon KatdAAnda yia va eifete Ou 1g Kade n UNAdPYEl TP@TOS p > N

Aơxrn 12 Na detéete bt ay p etvai mpdtos tote tte p = 3 (mod 4) cí£ p = 1 (mod 4)

Aơxrnơn 13 Na betfete 6t1 Undpyouy drepoi Teatoi tNS popens 4n + 3

Aơxrơn 14 Na ôcfct< bu av p etvai mpdtos tote tte p = 1 (mod 6) efte p= 5 (mod 6)

Aoxnon 15 Na detfete 6t1 undpyouy dneipoi Tpa@toi TNS opens 6n + 5

Aoxnon 16 Na deifete 6t1 av n etvai DetiKds axépaiog peyadUvtepos and 1 Kai n|(n —1)!+1 téte on efvar mpdtos aoiHóc

Aơxriơn 17 Na detéete dt av mlab Kai (m, a) = 1 téte mI

Aoxynon 18 Na detéete dti av (m,n) = 1 tote

(mn, a) =1 av Kat pdvo av (m,a) =1 Kai (n,a) =1

Aơxrnơn 19 Na detfete 6t1 to Adyioto Kowd toAAaTAdoio bUo DetiKOY aKEepatay

mn

m,n €tvai too He (m,n):

m,n

1.2 AtueĂAecfc Iloở€etc Ilœoœðefyueœrœ AĂYeBet<¿v

Aouov

Mia ðywẽXÍc noởŠn drĂó u(v mean of Eva ovvoko X etvou ula ouvdotnon

f: XxX + X, ve Dra Adyia Evac xavdvac f y~ow tou onoiou oe ono1odr-

note Cevyder (x,y) and otoryeia tov ouvddou X avttototyouus Eva teito Tou TO

ouuBoriCouue ue f(x,y) Luvidas 7 ovvdetynon f ovuuBoriCetar ue øÚUÖoĂđ óc +, :,* X0 Ypđ(oUue + + œvtÍ +(Z, 9)

Mia adyeBerxy Soy anoteAcita and Eva obvoko X xou xdmoves TEdEEl¢ TOU EXOUV optơtẻ øe AUTO “Evac and touc oxomotcs trục đĂy£edœc sivon va xataTd&€er xơt vd ucĂctfloet tic adyeBoixĩc SouEec avddhoya ue yevinec WWLOTHTES TOU UTOpEt

va ĩyouv ot Tod€etc Lav nrapdderyua avapgepouue thy xoóơÐcoør) dxĩpdteV dovD-

UV, THY toóơỦcơr) ồt%vuouớtoV trì TEdoVEoN ouvaeThoewy Av xat eivat EVTEAWS Siapopetixes Tod€eic oe Siapopetix& ovUvoAa yolpdCovta xdmoles xolves LOL6THTEC OTNWS 1 NOOCETALOLOTIXOTHTA , n UNAEEY OUdĩTEpOU oOTOLyEloU xAT Eival OUVETOS Xcfotuo vơ Yvoof€ouUe TL OUUTEEdoUATA UTOpEl va TEOKUPOUV UE LOVY TANEO- (0oofœ tồtótritec trịc TEGENS xa ac UN YvwpiGouue thy fdia thy TEdEN WotTE va UH yperaCetar va anoderxvvouue of xaVe Eeywoioty adyeBorxy Sour ta Sia noớyuo-

tơ Oa Eextvioouue ty xatataen tov aryeBoixav Sou@V aNd Tic TIO aTAES (xa TIO YEVIXES) MOC TI¢ TIO GØÚVỦE£tEC U£ XÚOt% avapOE& OTHY Evvola Tou dSaxTLAIOU tou Da opicouue Atyo TapaxdTH

Optonds 13

Hynopdĩa ctvai ĩva atvodo X oto onoto ĩxeat opiotel pia dyeAS mpocetaipi- oukn Todĩn *, pe dAAa Adyia Evel opiotet pia ovvdptnon *: X x X > X wote Via onoiadnrote ZY, 2 € X wyvel:

‘Eva dddo xœođồetyuo ritouớồœc (unị avtietadetixtic) efvoar to ø0voĂo XÊ óĂœv

16V ƠUV0otflOeov ƒ : X —> XÊ and Eva obvodo X otov eautd tou UE THY TOGEN THC ouvvEons GUVAETIOEWY

Ocuouóec 14

ïXonotuoroteÍxơt xo o ópoc œ3£Ătœv1

Opdéa ctvai €va aotvodo X ato onoto éyvet opiotet puta dyseAHS Tpd&y * nov éyel TIS TAPAKATH 1016TNTES:

Tìa onotdỗfrot< #,t,z € X,x*«(y*z) = (ay) #2 (1)

Yrdpyaa e € X dote pra Kdde rE X, wyvern*e=ex*L =X (2) Ita kúÖc x © X undpyet (uovadixd) ye X, dotexxy=yxx=e (3)

To ơtoweío e otny napandva e&tawon (2) €tvai povadixd Kai Aéyetai ovdétEpo otoreto TNS TPdENS

Entons to ơtoryeío y otny napandva <Etowon (3) etvai provadikd yra doopévo Z, Aéyetai b€ avtiotpogo otoweio Tov x

llœodồefyudtœ ouớồœ eivat ta cvvoAa Z, Q, R tav axcpatwv, pNTOV, TOXYLATLXAOY goruay avtiotolya ws THY TEden thc TedoVEONC “Eva dddo Tapdderyya ouddac

(un avtivetadetixt¢) eivar to cbvoAko X* éhwv twv ouvaptioewy ƒ : X > X and

ÉVŒ GÚVOÀO X otov eautd tou Tou efWơt 1-1 xo ent, ue THY TEdEN trịc øÚVỦ£OIG ouvapTyoewyv Eda to oudétepo otolyelo Eval n TAUTOTIXY, ATELKOVION KAL TO a vríGteo0o ơøtotyefÍo uÍœc cuvdetnons f : X > X civar n avtiotpoyn ouvdetnon Ƒ}':X—¬X

Av * eivar wa Teden optouévn of Eva ouvoAo X, ÉVớ UnooÚVoÀO Á toU Ä AEYETAL RACLGTO Wo TEOS THY TOGEN auTY av yia onotadynote a,b € A ouuBatver axbe A Le auth thy nepintworn unopobue va opioouue ula Teden * oto A TOU civat O TEELOPLOUds THS œoyt⁄Íc noớệnc oto A Av to Cevydor (X, *) Eyer dour

nutouddac (avtiotorya, ouddSac) téte Eva un-xevd Unoobvodo A tou X Va AEyetaut UEO-NLLOLASa (AVTIOTOLYA UTMO-OL.GSA) AV Elva XAELOTO WS TEOS THY TEGEN Xơt

to (A, +) éyet ồot nutouéđồœc (œvtfototyd, ouddac)

l[lœozrrieoÓue órt ytœ VŒ cÍVơt vớ urI-X⁄eVÓó UtYOOÓVOÀO A utớc r)utooUớồớœc UTO- rnutouớồ% noéret Xơt doXeÍ vớ eivar xACloTO Wo TEOS THY TEdEN Aut ồev civơt QOXETO GUWCS OTHV TEpiTTWOT Tov oUddwv OuuiTouue dt of AUTH THY x€ÐÍtt@OT) Eva UN-xXEvVO UTOOUVOAO A Ula oUddac eivor UTOOUdSa av xan UdVO av yla xaDE a,b € Ataatb,—a,—b € A Avédoyes Evvorec uno-douav (Um0-SaxTVALOI,

UNO-OMYATA XAT) Va optotoby TapaKdTw

1.3 H évvoww tou SaxtvAtov Opiowdc, ueotxéc Bơ-

OLKES LOLOTHTES xXaL TaPAdElyUaTa HỈ ¿vvotœ TOU UTO-SAXTUALODL

H gróuevn Sour tou Va opicouyue éyet ồÓO nođ€£tc Xơt sivon auty Tou Va anoteA€éoe!

TO XUELO AVTLXEiUEVO THC UEAETIC WAC

Ocuouóec 15

AgakcúĂtocŠ <tvai ĩva otvodo R ato onoto ĩyouv opiotel S00 Sipedets mpd&ers +, - MOU EYEl TIC TapaKdtw 1016tNTES:

To (R, +) efvai avtipetabetixh opdda pe ovdĩtepo otoryeto to 0 (1)

H rpdĩn- etvai enyepiotiKh was mpos thy + dndadh

yia onoiadnrote 2, y,z € KR tơxúÚ«

Ocuouóec 16

Oa ovoudlovupe ĩva ồakvúĂo (R, +, -):

(a) Mn-trctptHiĩvo, ga ĩvei toUĂóvIơtoy ôÚo otonyeta

(6) AvrperabetiKd, av dvo onoiadrnote Øtoiv€Í4 toU d,Ù g/tiuictdtfD€Vtd1 ws TOOS tHv -, OnAadh ab = ba

(y) AaxtitA10 pe povdĩa, av n nuoudda (R,-) Ever ovdĩtepo aotoiyeto, SnAadr umdpyel €va lp € R pe thy id16tnta yia orowdhrote a € KR va oupPatvar lpa =

alr = a

(5) Ay (R,+,-) cai ồaktúĂioe pe povdda 1p, Oa ovopidCovpe ĩva otoryeto tou a aytiotpĩeynipo ay utdpyei kdroi b € R pre ab = ba = 1

Ot nodgeig +,- Tov opiGouue oe Eva SaxtUALo oUUBOAITouv apnonueves TEdceEtc

xa dyt Tic ouvnDiouEvEs TEdEEIc THC TEdOVEONS xa TOU TOAAAaTAACLAOUOU Qotd-

oo Va ovoudTouue ovyvad THY <+» TEdoVEON Xa THY «-» TOAAATAQOLAOO

To ouồếtepo ototysio thc (R, +) ovuyBoriZeto ue 0 xa to avtioteowo Evdc otol-

yetou © ws TOG THY TEGEN + ue —z Emfone Ủœ Yeđ@ouue # — œvvÍ toU #-} (—9)

‘Evacg roĂÚ0 yefOtuoc øUuuÐoĂt0uóc eÍVơt 0 Tdo0XZớtG) :

Ocuouóec 17 Av A,B etvai ðúo onotdðfrot< HIỊ Ke#ú UROƠ0ÚV0Ăđ cVóc ồqdKtUĂfÍo0U

O rapandve ootouóc etoớyet ồÚo xođŠetc oto øÚvoöÀo 2 óÀœV twv UROOUVOAWY cvóc ồœxxtUÀÍou Jề Iloéxet vœ ndoœtrief{oouue ớt! to 2“ ồev cÍVdt ồotÚÀtoc uUe dUtếc Tic noớ£etc, œ0oÚ yYtœ ndođồetyud to (2%,+) anéye: nord and to va etvor oudda TIapatnpetote wotdoo ot: Va Eyouue navtote A+ B= B+A yratt yn TodEN + cfvơt avtiuetavetixn oc xaVe SaxTVALO, Cyt GUwWS anapattnta A- B= B- A, extdc

xơt av o SaxtvAtog eivat avtiusetavETlxdc

Tlapdberypal Av pe (Z,+4+,-), (Q+,-), (R+,-), (C+,-) cupBodtooupe ta

ovvoAa tov akepatwy, pntay, TpaypatiKay, pryadikady api0pay, aytiototya, PE TIS ouyvnthopéves modfeis, autd anoteAoty napadefypata daxtvAtwy Eidixd avtd etvat Kal nmapadetyiata avtipetabetixay OaxtvAtwy pre povdda

IIœoởðctyuœ 2 Tơ X €va onoiwdrnote in Kevd atvodo Kai R to đÚyoÀo óÀú/

twv utoouvdAwy tov to R optloupe dv0 mpdéets:

Oi mpdfeas ato bettepo péAos tay rapandyw eliadoewy evvoettai 6t1 €tval o1 Ov- 1e noáÈe o10 Z Tia e€doxnon ypdgoupe toy rivaxa tov mpdfewy ato Ze Kat aprvoupe pepikd Kevd va ta ovpTAnpaoe 0 avayyaotns

f+g(x) = f(z) +9(2) f-g(x) = f(z) - g(a)

To (R*,+,-) <tvai daxttAros

Tlapdberypa 5 Eotw (R,+,-) évas daxtvAtos, n é/ac Ö€tIKÓC axépaios Kai éotw

I, = {1, ,n} Me M,(R) 0a cupPortCovpe to ƠÚ/0ÀO ÓÀ@V tay ameiKovioe@y filn x In 7 R Ta otorneta tov M,(R) 8a ta ovopdoupe n0 X n— TÍVGK€G 1I€ otoweta and to daxttA00 R

SuvHOas yodpovpe ay; avtt tov f(i,j) Kat avtf tns auvdptnons f yodpovupe (ai;) 1} Kat OAo toy mivaka tay otoetwy Eto yia mapddetyya o mtvaxas

rapiotdve: to napaxdtw otoreto f tov Ma(Zs): f(,1) = aun = 0, f(1,2) =

iC ) 1 avi=j

?; —

Tevixd o M,(R) dev etvar avtysetabetiKds daxtvAws axdpa kai avo R etvai Eva tétơto rapđồcpHa <tvai o1 2 x 2 nivakes oto R

Oa ypnoworoiouue cuyva xo Tov mapaxdtw OUUĐOÀIOUÓ Av ay, ,dn © R DETOUUE

Optopdg 18 AyneZKraace R Bétovpe

(—a) + -+(T—a) (nT— gopés) avn <0

Ayn > 0 téte optlovpe

a’ =G@:Q-a :-a, N— popes

Ot Baoixés wrdtyH TES tov SaxtvAlwy GUvOiTovTaL OTHY TAPAKdTW TEdTAON:

IIpétacn 3 Hota R évas daxtvAios n,m pvoikot apiOyuot Kai a, b,c otorneta tou

(8) a?*?2 = a" am,

(9) Avo daxttAios éxer povdda Kai rapandye and éva otoryeta tócc Ủd évoupe

3 (—a)(—b) = —(a(-b)) = —(—(ab)) = ab

4 Oewpobye Eva m otadeed xo epapudTouue exaywyy, oto n Tran = 1 7

oxéon (4) civat 7

(Q1 + +++ +Qm)bm = 0101 + +++ + Omby

HAL loYVEL AOYW THC ETIUEOtOTUXCÍC tỒtĨtYitŒC TOU NOAAATAAGLAGUOD WS TEOS THY TOOOVEOCN

Ag unxoVéoouue Tapa OTL YvwelTouUE THhy Loyd THC oyéonc yra SoouEVa m, N,

ồnÀdồ' Yytd oxotdồfrote ứi, ; đm, Ơ1, , Dạ € TÈ Yve@p(EoUue ĩtt toyÚet

10 Ac uxoÙéoouue ĩtt YtŒ% to a UNdeyE! Eva otOLycio 6 Ue ab = ba = 1g Tote

œV YtŒ% XớTOtŒ% #,1 ouuBatver az = ay Va Eyouue Ott bax = bay doa xa x = y AUrĩ ồefyVet Xơt trị uovdồtxĩtritd 1oU 0VttGtoĨ00U, apo Œv Uxfioy€ XớnotO Ù' ue

ab’ = 1p, tote ab = ab! xm and THY neoriyoÚUevr| xdodrfprior) Ủd cydue Tóc b=Ù O

Oa Unopoboe xdnotog va avapwTnvel av n Evvota tou SaxtuAtou UTopEt va Y€- VIXEUTEL APALPWYTAS THY araitnorn oO SaxtUALog va civan aBertavy oudda wo TOC THY TECOVEOY AAG anA& wa oUdSa Qotdco, av o SaxtUAtog Eyet Uovdda, 7 tSto YN) ETIUEPLOTIXOTYTA TOV TOdEEwWY anaitel THY avtiVEeTAVETIXOTYTA THS TEdOVEONG Todyuatt,ag unoVéoouue Ótt tGyYÚOUV ÓÀđ Tớ GỆt@U@t0đ Tou SaxTUALoU ExTO¢ ANd

to <a+b= 6+ a» xa ag TépoUUE SLO oxotdồfitote OTOLyela a, b evdc SaxtUAtoU

Lyedsdyv dUEGO ATS TOV OPLOUO Eivat to E€fc:

IIpétacn 4 Eva pn xevd unoatvvodo A evds dbaxtvatou efvai unodaxttAtos tov R

av Kai jidvo av ioxtovy ta €fhS:

1 To A etvai KAcioté ws mpos tis diapopés SnAadH av a,b € A tétve a—b = a+(—b) EA

2 To A efvai KAciot6 ws Tpos ta pivdpeva, 6nAadH av a,b € A tote ab E A Arddety: Ac unodéoouue ótt tơyÓoUV ta (1), (2) Téte Dewew a € A (aod éyw unovéoet 6tt to A efvar UN xevó obvoAo) Adyw tov (1),0=a-—ae A Extong ava € A tote apo 0E A,0-a=—ae A

Toc av a,b € A téte —b € A xt ovvenme¢ a+b =a-—(—b) € A xm abe A

‘Aoa to (A, +, -) etvar SaxtvAtoc

Avtioteowa, av unodéoouue ott to (A,+4, -) etvor SaxtvAtog Téte to (A, +) Da etvat oudda Av e civat to ovdétepo ototyeto tH¢ ouddac autyc Va Eyouue e+e =e

xa ouvemas € = 0 Auté duws onuatver 6tt to —a Va etvat to avtioteogo ototyeio

tou a otny (A, +) xo øuvext@ec —ø € A Exetồf and thy uxddeor pac to (A, +4, -)

elvar SaxtbAtog Va Eyouue dtt to A elvan XAELOTO WE TOS TIC TEGEELC KAL DUVETS

ava,be€ Ra éyouue ab €C R,a+(—b) € R xo detEaue dtr txavororovvtat ot (1), (2) O

Ac unovĩoouue 611 A etvoar Eva obvoAO and UTOOUVOAa Evdc ouveAoU R H

tony ()A drwv tov ototyetwv tou A opiZetat oav 1o GÚVOĂO ÓĂ@)V 16V ƠtOtyẺ@V

tou R nov avyixouv oe xaV_e ototyeio tou A,

zeƒÌì4 & VAEA TEA

xar ouuBoAiCetoat we (A

Ilapatypetote ott av to A xeotĨy€t texeodœoUĩvo xĂf{Öoc ơtotyeÍ@v ty v = {A1, , An} tote T0 ()A = AL MN -M An O ouuBoĂtouóc autd¢ slokyeta yra va opioouue

tr 1oMf{ «orxotoUðfxote xĂfÖouc øUvóĂov> Doớoouue mon xœ (]{A: A € A}

œvxí (Ì.A

TvŒc tooồÚvduoc toóroc vớ ÔĂĩnoUUE ula OVAAOYH ATG UTODbVOAa Evd¢ OU-

vohou R eivat oav wa oixoyĩvera (A;)icr and Unoobvoka tou R Sndradh oav uta ouvdetyon tou os xaV_e ototyeio evdc Soouevou ouveAou I avttototyel Eva obvoAO

ie

Me œutó tov ø0WÖoĂtOUÓ TỊ TOUWf| TWY OTOLYELWY TS OLXOYEVELAC Elva TO ØÚVOĂO:

(Ai = {zx: ya wade i, x € A;}

ArnddetEy: Ac unovĩoouue ott A civar Eva obvoAo UodaxtuAiwy To (}A civ

un xevd apod neptĩyet to 0 Ava,b ef )A tote a,b € S ya xade S € A xm œ0oÓ xớÐc ototysio S thc A civar unodaxtuAtog Da Eyouue Ott yra xớÙc 6 € A

a—b,abe S da a—b,ab € ()A xu to (\A Va ecivor uxodaxt0Atog AdyYH THC Ileórxdørne 4

2 Âc UnoÙỦĩoouue ótt , cÑớt to GÓVOĂO ÓĂ@V Twv UTodSaxTUAIwy tou R nou

neptĩyouv to A To A eivor un xevd agod Re A Apxet va xớcoute (4) = Í}.A HAL VX EYAPUOGOUUE TO TOWTO uĩ@oc xrịc lleóteonc LÌ

hav ĩva tapaderyua utodaxtuAiou tou Z avaggeouue CAA ta UTOGUVOAG TOU

tục uopgfc kZ = {kn:n € Z}, yra ĩva k ồoouĩvol!

1! Mepuxo( ơuYYedoẻc, ómoc oi S Mac Lane xu G Birkhoff oto noẲ YVeơtó ØóYY@duớ touc

«Algebra», otov optoud tou SaxtvAtou amaitobyv va ĩyet Hovớồo AvtÍØtotXŒ% G1OV O0LOUÓ TOU uto-SaxtvAlou aMaTOUY EMIMAEOY va TEpLEXEL THY Yovada tov SaxtvAtov Efvor gavepd mw ye autĩv tov optoud to Z sev ĩyet xaveva Yvhoto UrỊ tetptuuĩvo UTO-SaXTVALO.

IIœođðctyudœ 6 As Gewprooupe évay onoiwdAnote daxtvAio R Kai éva otoryeto tou a And tnv Il[pdétaon 5 Oa undpye évas Adyiotos unodaKtvAios tou R rou

Ja repiéyvet to a, mov tov avpPodtCoupe pie (a) O daxtvaAtos autdés ba repiéyet avayKaotikd ta a, —a,0 adAdd kai Aa ta abpotopatd tous Kai yivdpeva tous Eto etvai evKoAo va dovpe 6t1 Ba nepiéyer KúÖc ơtoty€Ío trịc Hop@f{S:

(a) = A

KÀeWouue thy ãœoớyed0o ue ÉVœv optOUó:

Oetouóc 20 Kévtpo evds ồaKtuÀfoU lì ovouúšoUbe to atvodo C(R) óÀ@w tay otonyelwy tov nov aytipetativeytat pie Aa ta undAoina AndAadr

C(R) ={ac R: Tìa kóÐc + € R,ax = xa}

ouot

H yevuxf évvotv evóc oeœtơto0 (Í oooedtouo0) ustagv S00 aryeBorxwv ồou@v X,Y' tou tồíoU túnoU (nutouớồec, ouớồcc, ồoœxxtÚÀtot, 0X⁄Épotec xeptoXéc, oMUaTa XAT ) Siveta oa Uta ouvéetnon f : X —> Y n onota ồtdtr)pef tíc avtiototyes Ted€erc KéeSixevovtac otyy Tepintworn twv Saxtudiwy Eyouue:

Ocuouóe 21

(a) Eotw R, R' ồaKtúÀiot wai f : R > R' Oa ovopdlovpe tny anexdvion

f opopopqiopnd SaxtvuAtwy 1} atAd opopoppio po ay ikavoroiel tig TapaKdta Ovo 1016TTES:

(1) Iva onoadrnote a,b € R, f(a+6) = f(a) + ƒ0)

12 Mia roÀ0 xdÀ4 đoxrjor) eÍíVơt với đœroồe(Eete te XớÖc ÀETtCOUÉOELO TOV TapATdvW LOYUELOLO , 2 2 + / z Zz z ⁄

2) Iva onoadjnote a,b € R, f(a-b) = f(a) - f(b)

H etxéva tov opopopgiopov f «fvai to avvodo:

Imf = f(R)={f(z): ce R}={yeR: Ave Rypcy= f(x}

O ruptvas tov opopoppiopov f etvai to ovvodo:

Ker ƒ = {z € F: ƒ(z) = 0}

(B) Áv é/as ouoHoo@iơuóc daxtvAtwy etvai 1-1 ovvdptnon Oa tov Aéue provo- popgio pnd SaxtvAioyr

(y) Ay €vas opopopgiopds f : R— R' daxtvAtoy etvai ovvdptnon ent tov R'

da toy ÀCH€ €Ttt-IOO@171LÓ ŠŒKCUÀ (01

(5) Tédos, av évas opopopgpiopids daxtvAtwy etvai 1-1 Kai ext ovvdptnon ba toy ÀCH€ 1701000171Ló daxtvdiwy Ay yia dvo daxtvdAtous R,S tvyatver va umdpyel ioopopgiapds and tov R otoy S ơi ồqKcÚÀtot Ba A€yoytai 1odpopepor Kai

da yodpoupe R= S

H endyevn medtaon etvor ula andy xor “aA choxNon ottc Evvolec TOU LOKIC OpfGdUe:

IIpétacn 6

Loto R,S daxtvaAwi Kai f : R + S €vasg opopopgpiapds Téte:

(a) H c<óva f(R) tov f efvat vnodaxtvAtos tou S Kato ruprvas Kerf tov f etvat umodaKtUAtos tou R

(B) Hf etvai hoVoHop@iơHóc av Kat pdvo av o muprvas tng €tvai toog pe {0} (y) Av o f etvai povopoppiopds téte o R etvat 1adpoppos pe tov UnoðaktúÀio f (PR) tou S

H oyéon R& S éyet ddec tic idtHtEs plac oyéong toodvvauiac'® Iloớyuơtt,

œV Tđ@OUUE To£tC OftOtOUOỒftote ồœxtÚÀtoUc T, 5, ' xót Va Eyouue Ott:

1 3ï

2 Av J3 Sxóte S5 % lì

ỏ Av 3 Sxơư S47 'tóyc R7

Tia va Sovue étt toyvet to (1) apxet va ã0odtr)ipf{OOUUE Ótt rỊ t0ŒUtOtUXf đAEL-

xdvion and éva SaxtUAlo OTov EaUTd Tou eíÍVơt tơotUoeootouóc Tia to (2) aoxet 1O YEYOVÓC Ótt TỊ ŒVtÍØtOO@T| 0£X⁄ÓVIOT) tGoMoO(@t0UOÚ 0Ó tov Í¿ ơtov S etvou tooUooo0touóc œró tov 5 otov P TÀoc to (3) npoxvnte: and to yeyovdc 6tt 7 GUVVEOT IOOMOPOLOUY €ÍVớt tGOUOOO@tOUỐCc

13 Acy uxoeoÓue turtt⁄ớ 9œ toÚue ótt rị & elvou pra oyéon tooduvautac yrot! Sev unOpOUE Va với xoœopÍGouue 1o obvoAo oto onolo opiCetar Oa UnopovoauE va TOUUE ÓtL Oo(É£T0L ØtO <GÚVOÀO hwy Tov SaxtvAlwvy wAAd HUT Sev elvan emitp|EeTT6 OTHY OVVOADVEWPLA, OTW?S AUTH AVATTUCOETAL OTO AELWUATIXNG OVOTHUA Twyv Zermelo-Fraenkel mov efvou xo to mio SadeSouevo Tia to Adyo œUtó øuvfÐo@c Àéuc ótt rị & elvou ayéon toodsuvautac othHy «xrAdcon» drwy twv SaxtvAtwy.

Abo toduop@ouc SaxtUAtouc axdua xa av eivar Stapopetixol cav obvoAa Da Dewoobue Ott TaUTITovtTaL UETAEU Tous and aryeBoerxy drovy Ilpđyudrt, oxotdồf- note adyeBorxy idtyTta Eyer Evac SaxtUAlog (MENEEMoUEVOc, œvttuetœÔettxóc, Éyết uovada, civar axéoaia Teployh, eivat OWA XAT) THY ia Ya Eyer xo onoLloodshnotE đÀÀoc ðœ⁄xÓÀtioc tøóUop0oc Ue aUTOV Me autt tH Aoyixh Va AEE TI:

7 Evas daxtvAws lÌ Tr€01CV€Cdt 100110001Kđ 0c 7 , ,

éva daxtutAio S ay etvai ioopoppiKds pe €va UT0- daxtUAio tou S - ff ioodvvapa - ay undpyel €vas povopoppiopds f: RS

Ocuouóe 22

Ileóxdơn 7

Kade ồaKtÚÀioc 1<p1Óv€t01 120H00@1Kđ 0c é€va ồdKtÚÀIo Hé Hovyđồa

AnoddetEn: ‘Eotw R évac SaxtvAtog xa Eotw S = RX Z “ito S optCouye ồÚo TOỚC£€tC

(ø, m) + (b,m) = (ø + b,m + mì) (ø, mm) - (b,n) = (ab + ma + nb, nam)

O § yWetơt ue qUtếc ttc modEetc SaxtvAtog ue Yovada lg = (0,1) xơt undév to 0s = (0,0) ónœ< cÚxoÀœ uxoooÓue vớ exdÀncÚooute ©e@ooÚUE trỊV 0etXÓVtGT

Ƒ:R— Sue ƒ(r) = (r,0), nov eÓxoÀœ ồeÍyvouue ótt eWVơt oUouoeotouóc ‘Apxet

va defouue ott civa 1-1 xan yt avTS to ƠØXOTXÓ €€£tớđCOUUWE TOV TUOfVƠ tíịc: Av f(r) = (0,0) téte neopavac r = 0 xa ouvenads Kerf = {0}, SnAadH n f efvou 1-1

O

IIœoởðctyuœ 7 EHotw R évas onowadrnote daxtvAios pe povdda 1p H areixé-

vion f :Z— R, nov opttetai ané tn oxéon f(n) = n1p, €fvai opopoppiapds

Aoxnon 24 Eotw R,S dto daxctAioi Kai oto R x S optCoupe (r,s) + (7’, 8’) = (r+r”,s+3),(r, s)(r', #) = (rr”,ss!) Enadndevoete 6t1 to otvoAo Rx S etvat da- ktúĂioc O ồaKtÚĂioc autds oupPodrtletai ue ROS Kai A€yetai to evdd đổootơ1td

tav dvto daxtidtoy R,S

Aoxnon 25 Ay R etvai ĩvas daxttAiog tĩte Kain € N, peyadvtepo ard 1, oto

ơÚyoĂo Từ" = lì x - x R optlovupe dv0 rpd&ess:

(đ, -; Xn) + (41, +++) Yn) — ((ì + y1),- -+5 (Ln + Yn))

ti cae En) (yi, cee Un) = ((z14), “ng (Znn))-

Na «&etdoete av o R” pe avtĩs tig modes efvai Saxtvatos

Aoxyon 26 (a) Eotw R ĩvas daxctvdios, a € R Kat

C(a) = {zr € R: ax = xa}

va detEete 6tt o C(a) efvai umodaKtvAtos tou R

(B) Av AC X kat optoovupe

C(A) = {z: yia kâÚc a € A,ax = xa}

va detEete 6t1 0 C(A) etvai unodaxtvAios tov R rates Kai 6tt C(A) = ¿a4 C9) Act&te axĩpa tas av o R etvai daxctAws pe povdda tote o C(R) etvai ayvtyeta- DetiKds SaKtvAios pe povdda

Aoxynorn 27 Eotw (G,+) pla aBcĂtaf{ opdda Kai End(G) to ovvodo (wy tay EvoopLoppiaopay tS opddas d6nAadh to ơÚ/oĂ0 CAwy twy aneixovioewy f : X 4 X

pe f(a +y) = f(x) + ƒ(u) yia onoaðfro +, € G To End(G) to epodidloupe

pe ðúo tpdĩers f +9, f-g dnov (f +g)(z) = f(x) + 9(2), f- g(x) = f(g(a)) Na detEete 6t1 to End(G) pe autĩs tig mpdĩ&eis efvair daxcvAtos

Aoxnon 28 Eotw C o daxttAtos tay pryadixay api0udy va «Cĩtacete noia ard

Ta TapaKdtw unoovvoad tov efvai uTodaKTUAiol

Aoxyon 30 Eotw M2(R) o daxttAis OAwy tov 2 x 2 mvdkav ndvw oe ĩva

a , ) € M;(R) ooffouuc det(+) = ad — be

Na det&ete bt det(xy) = det(x)det(y)

daxttAio R Tia ĩva otoryeto x =

Aoxyon 31 Eotw M2(Z,) 0 daxttAws CAwy tov 2x2 mvdKayv tdvw oto Ly, 6TOv

Jf, # Z, z ⁄ ⁄ —— a b Jf, ⁄

0 p cíW/aI nodtoc Na dbetéete 6t1 Eva otoryeto tov A = o gq } Sư ay tio THEY pO

av Kai pidvo av n «optlovad» tov det(A) = ad — bc etvai diagopetixh and 0

Aoxnon 32 Av o R etvai petatetixds ðaKtÚÀioc Kơi a € R va detéete 6t1 TO otvoko A= {xz € R: ax = 0} efvai unodaxtvAws tov R

Aoxnon 33 Eva ototyeto a evds daxtudAtou Ba A€éyetai évas ồtŒIOCTỊG C0U LTỊ- devds ava #0 kai undpyei Kdroi b € R pe b #0 Karab=0 7 ba =0 A vnapén ƠIữI0€t@V tơU pindevds oc Eva daxtvAw onuatvel 6 bev 10XÚ<t 0 axdAovOoS Y6pLOS

«abị => à=0 f0 =0»

Na Bpette tous diaipétes tov pundevds atous napaxdtw daxtvAtous: Ze, Z7, Ze

Aoxnorn 34 Av a,b elvai dv0 otoneta evds daxtvAtou nov aytipetatiberta va d€tSETE OT!

(arora ars (Sartor + (Pantha bo

OTOU

n\ _— nÌ ke

bk) Ein—k

Opiopds 23 Eva ototryeio a evds daxtvAtou (R,+,-) Oa Aéyetat

(a) tavrodivapo ay a*® = a (8) pnbdevodtvapo av yia Kdnoto axépaio n > 0, a” = 0

Aoxynon 35 (a) Acéte 6t1 éva pin pndemKé tavtodtvapio otoreto evds daxtuAtou dey efvar pndevodvyvapo

(B) Act&te 6ti KdDE pin pandeviKd pndevodvvapo atoryeto etvai Hiapétns tov pnde-

ĩc

(y) Na Bpeft ta tauuoồÚvdua Kai pndevodtvaya otoryeta oto Zs

(ồ) Na ồcÍctœ ĩtt to Hĩyo HnồcVoồÚv/dIHo otoryeto putas axépaias tepioyrs €ÍV01 to

0 Kai to pdvo pin pundevixd tavtodvvapo n povdoa

Aơxrnơn 36 to ovvodo tay axepatwy Z x Z optCoupe O00 mpd&erw pe

(m.n) + (m',n') = (m + mm”, + n)

(m.n) - (mĩ, n') = (mm, nn!)

Na delete ớt to 2x 2 pre autés tig mpd&ers efvai daxtvaAws pe povdda Kai ti to

Z x {0} efvai vnodaxctAtos tov pe povdda entons tov duaws «tvai diapopetiKh and qUtf⁄ toU 2 x 22

Aoxnon 37 O opiopds tou unodaxtvatou dey anaitel va Ever povdda ay o daKtU- Àtoc évet MnopeÍ va npoKÚỦoUV didpopes nepintaéaeis Om@s:

(a) O ðaktúÀioe va tnv vét Hovúồa qÀÀá UroồqKCÚÀloc va ÓV€l

(B) O vmodaxttAtos va éyer povdda Stapopetixr and avtr) tov daxtvAtov

Na detéete 6t1 av ovpPatver pita and tis napandvw mepintmceis tote Nn povdda tou unodaxtuAtou evar diaipétns tou 0 Hpoonatetote va d@oete napadefypata tou det- XVOUV ỐtL 01 Tponyovpeves Kataatdoets pitopel va npoKÚỦoUV (dette Kal THY TOON- youuevn doxnon)

Aoxnorn 38 Av R etvai évas daxtUAtog pe povdéa 1 Kai to a etvai Eva pndevodv- vapo otoyeto tov téte to 1+a etvai avtiotpéynpo (Ynddaén: Av a” = 0 Bewpetote

tol—a+a?— (—1)""'a"")

Aoxnon 39 Na detfete Út! to KÓ/tpo cóc ð0KtUÀfOU Etval UTOdAKTUAIOS TOU Aơxrn 40 Eva otoryeto a pitas nuiopddas (H, *) 0a Aéyetai tavtodivapo ay a*a=a To ovdétepo otoryeto putas nuiouddas, av undpyel, «fvai tautodvvapo

As unotéooupe tépa ti to H etvai menepapévo ơÚvoÀo Na detéete 6t1 utdpyel pila touddyiotoy Adyiotn vTo-nuiopdda ths H SnAadr éva pn Kevd vnoatvodo A tou H Kdeiotd ws tTpos thy Tedén * Kal TéTOIO Mate Kavéva yyrj}o10 UTfO0Ú/0ÀO tou dev efvai KAElotd wS Tpos THY Tpdgn Ltn ovvéyeia va detfete 6t1 to A Oa etval avayKaotikd povoovvoro Kai va ovpmepdvete Oui KdV TEenNEpaopEevn nuiopdda Mpéenel Va ével taUtoðÚVdHO Ơ0toiv€fo

Aoxnon 41 Evas daxttaAios R Aéyetai 6axtuAiog tov Boole ay éyei ovdda Kai Kd0e otoryeto tov elvai tautodvvapyo ws Tpos tov moAAanAaoiaopd Na detfete Ớt! éVữG tÉtoloS€ daKtUAlos efval avayKaouikd aytipetavetiKds

1.5 HH évvotœ tn< XŒ@%tr)ptƠttXfGC evdc dSaxtvAtou

2;U0VeyÍCoUue Ue Ula aXdUN YEHoWWN Evvora Tov efvat n ÉVVOtO TYỊC X@@dXtTIOtOtUXCfÍC evdc SaxtuAtou

Ocvuouóe 24

Eotw R daxtvAios Av urdpyet évas avotnpd ÖctKÓC axépaiog n pe thy 1016-

thta na = 0 ya AAa ta a © R téte o juxpdtepos guoikds n © N \ {0} pe tny

lỒtớtrira 1q onoioðfrote a C R va ioxyver na = 0 A€yetal Nn YapaxtnpiotiKT tov daxtvdAiov Kai ovpPodtletai pe x(R)

Ay dev undpyet vas tétoios puoixds n Oa A€ue Ot 0 baxtUAlos éyel yapaKtn- ptơtiKf{ 0 kai 0a ovpBodtCoupe x(R) = 0

Uyyelwon 1 Ói ôakrúÀioi Z, R, C évoU dAdo yapaxtnpiouKh ton pe Of E- vas meTEpacpéevos daxttAlos dev pinopel va éyvet yapaxtnpiouKh ton pe 0 “Evas daxttAios Oa éxel YapaxtnpiotiKh ton pe 1 av Kai pidvo ay elvai TETpIPLEVOS

4Mepixnol ovyypagetc Vétouy yapoxtnpLotixy fon we oo avtt 0

Eivat euxoAo va Sovue OT loyvEL N TapAxdtw TEdTACN:

IIpétacn 8

Eota R évas pn tetpipévos daxtvtAios pre provdda pre yapaxtnpiotkn diagope- uukn and 0 Téte n yapaxtnpiotKh tou efvai o pixpdtepos puoiKds api0uds n pe thy 1016tHta N1Rp = 0

Amrỏðgtcn: Ác unoVecouue dott k elvan o uixedteeog 0@U0tXÓC dOtÙUỐC n UE THY tồtótrtd 31p = 0 Ilapatnoowue ott yra xdde a C KR, n Va éyouue tt na = (n1lp)a, ouverts ka = 0 xa ouveras x(R) < k AMadl = x(R) < k Sev unopet va ouuBatver yratt tote 11 = 0 xơt n modtaon anodetytyxe LI

Aoxnon 42 Aciéte bt évas OaxtvAtos pe povdda ével yapaxtnpiouKn ton pe 0

av Kal pidvo ay Tepiéyel 1aouopgiKd to baxtUAIo twy axepator

Aoxynor 43 Na unodoytoete thy yapaxtnpiouKh toy napaxkdtw daxtvAtwy: (a)

2222, (B)Z17, (y) C

VW ULATOL

DE AUTH THY mdođyeœ0o Va xdvouuE Ua TEATH Ta€woulon xaTHyYooIMV SaxTUAl@v

(R,+,-) xupiws o€ oyéon ye tic idtH TES THC MOAAATAMOLAOTIXYS NULoUdSac (R, -)

Ac Dewprfoouue Eva un tetotuuévo dSaxtbAto (R,+,-) ue yovdda xa tHY tị

uioudda (R,-) Méyor tapa dev éyouue ovCythoet tinota oyetix& we auTH THY

Nut-oudda Auté nou Brkémouue auéows civar ott dev unopet NOTE va etvor oudda agou to 0 dev unopet va Eye avtioteo~o

Ocuouóe 25

Ta 0totyeÍo a cwóc ðaKtuUÀfíoU R Hé Hovyđồa ln Đa Àéy€tứ1 ŒVEtƠtCDCỦ1ILO d Uápx«t éva ơtoryefío b € R pe ab = ba = 1 To otoryeto avté -ay undpyei - €Í⁄Œ1 povadiKd Kai ovpBodrtteta pe a* To atvodo Aw tay avtiotpeynpay otonetov tou R da ovpPodtletai pe R*

Tia napdderyua oto Z to uóvo œvrtơto€tUo ơtotyeÍo civ to 1

TlapdSetypa 8 Eotw X éva pn xevd atvodo Kai R* o ồqKCÚÀio€ ÓÀ@# tá ov- vaptnoewy f : X > R pe tig ovyntiopéves mpdfeis tov abpotapiatos Kai yivopévou auvaptnocwyv To 0 tov daxtvatou efvai n tavtotikd 0 ovvdptnon Kai n povdda tou eval n ovvdptnon nov rafpyver raytov thy tyin 1 Me avutés tig mpdéeiy puta

f €R* chai avnotwpedapn av Kai pdvo av ya wdbe cw € X f(x) £0

Ocuouóe 26

‘Evas daxtvAtos pe povdda otov onoto Kdbe pin pundemKo otoryeto aytiotpépetat

@s mpos tov toAAarAaoiaopd Oa Aéyetai daxttAtog Siaipeons Av eninA€éoy o daxtAios diatpeons etvai avtipetabetiKds Oa tov ovopdlovpe Eva Opa

Ltoucg XÀđøtX⁄OÚc SaxtuAtoug aprOuav yvwelTouue Sti av To ytvéuevo Sv0 d-

@tuœv efvor undév tote Evac and touc dvo Va etvar Auth 1 wordt ta efvo TOAD Yefowy xu THY yeNoworolovuE TY Yla va AUVOUUE E<iawceic AUTO dum Sev ouuBatver yewixd oe Saxtudtouc ‘Eto, dev etvor Sboxoko va Boowue duo trị tr)ồc- VIXOÚC TEtOOY@)VIXOÚC TIVAKES TOU TO YlvOUEVO TOUS va Elva o uNndewixds TivaKxac

7, 860 un unồevtxéc TENYUATIKES GUVAPTHOEIS TOU TO YIVÓUMEVÓ TOUC Vớ EÍVớL TỊ tr devixh øuvớotriorn, xox Lœ vœ cŠctớ0oUUE tếtotOU gÍỒOUC TEOLrtG@OEIC EtGớYOUUE TOV TŒOđXớt@) OOLOUÓ:

Ocuouỏe 28 Eotw R daxttAiog xaia,b € R pe a,b #0 Av ovppPatver ab = 0

ta a,b ba ovoudloyvtai b1aipéteg tov pndevdsg 2/UYK€KDIHHÓVG to a Oa A€éyetat apiotepds O1aipétys tov pndervds kai to b beE1d¢ S1aypétns tov pndevdc

Tlapdberypa 9 Eva yapaxtnpioté rapdderyya daxtvudAtov (nov pdAiota etvai a- ytipetabetixds pe povdda) mov éyet diaipétes tov pndevds etvai o Ze agov 2-3 = 0 Etvai pavepd 6u Kdbe daxttaAtos ths popens Z, pe tov k va puny etvai mpatos api0- pds Oa éyver Oiaipétes tov pndevds Ipdyyati, av ok dev etvai mpwtos tote ypdgetat

oav k = kiko, pel < ky, ko < k Otky, ko © Zy wat kyko = 0 Avtiotpoga, av ok

etvai Tpa@tos dev propel va évet Oiapétes tov pndevds [pdyyati av avté ovvéPaive

da unjpyay O00 0 < 1,97 < k tétow dote 17 = 0 oto Z, SnAadH o k Oa diaipovoe Suvetas ok Ga dtapovoe tov 2 efte tov 7 ÂUtó dpiws etvai dtoro apov o1 apiOyot 2,3 evar Detixol axépaioi avatnpd juxpdtepoi and tov k

Ot avtwetavetixot SaxtvAtot ue Yovada ywpic ồtdtoÉtec xoU unồevóc anotehovy uÍœ øntœvxuxf xdrryopfœ ồœxxUÀÍ@v xơt Ủđ touc SwcouuE Eva Ceywolotd dvoud.

Arnddeéy: H oyéon ab = ac civ toodbvaun ue tH a(b — c) = 0 xo ouvermc (aod dev Eyouue dStatpétec tou 0) Da toyvet av xa udvo av a # 0 cite b—c=0

H roatyn nepintwon anoxdetetar and undveon doa Va toyver n Sevtepn, SHAadH b=c

IIpétacn 10

H yapaxtnpiotixn pitas arépaias nepioyns R «tvai ette 0 ette mpa@tos apiOyuds

Anddegy: Ac urodeoouue Ott n yaouxTHELloTIXh THC axEpatacg mEployAc A THC orofœc trị uovớồœ øuuÐoÀfŠouue uơ 14 civat fon ue rn > 0 H yaoaxtrerotixy utoic 0⁄ÉooœtŒc TEeployyc Sev uTopEt va elvan fon ue 1 yrati ot axépatec meployéc Sev civ TETOtUUÉvot Saxtudtor ouvetac Va etvor xdmotog aprOudc n ueydÀÚt£ooc TẢ ÍGoc xoU

2 Ac unoVécouue Aotndv 6tt to n Sev Elvan TeE@tOG ONdTE Va cuUBatver Nn = 1 No

ue 1 < m,N2 <n AhA& tétE Va elyaue 611 N1y = Ni14+ Nel, CUVEerac Va ouveBaive M14 = 0 H nel, = 0 xm and thy noonyouuEvH Ilodtaon 8 Va efyaue OTL YAoUXTNELOTIXYA Da Atav 2; 4 Ne, dtono OO

‘Eva Queso TOPlOUa THC TEdTAGNS aUTIC Elva 7

IIpétacy 11

Kdde renepacpévn axépaia mepioyn éyet yapaxtnpiotixh ton pe Tpoto apiyd

Txớeyouv ðœxxÚÀtot ðtdÍo@cơrnc mou dev civ omyata “Eva onuavtxd mad Seryua UN aAvtiuETAvETLXOU SaxTUALoU Siafoeanc civan or «teteddec tou Hamilton» rou Da SobUE AUEGWS TACAXATO

Tlapdberypa 10 O21 tetpddes (4 tetpdvia) tov Hamilton

Ag éexivrooupe urevOuptCovtas tov opiapd tay juyadikay apitpudy To ovvoAo C piyadikady apiudy tavtidetai pe to oUvOAO RXR OAwy tay Cevyapiay TpaypatiKay ap1HMd/ ơto aroto évoupe optoe: Ovo mpders:

(a1, G2) + (b1, b2) = (a1 + b1, a2 + ba)

(G1, G2) - (bạ, bạ) = (ab, — agbe, ayb2 + aod)

2;Uf†Öœ to Cevydpt (a1, b,) to ypdgoupe cay ay1+aei é6nov 1 = (0,1) xari = (1,0) Kai 0 toAAatAaoiaopds Kaboptletai povoonpayta and tous Kavdves:

Me autés tis mpdéeis o1 duddes tpaypatikady apipay ytvoytai €va oa@pa, nov etvat

TO YYWOTS HAS OHpa tov pryadiKay apiOudy

O William Rowan Hamilton avaxdAvuwe vú tpoóno va optoer b0o0 mpdéeis pe- taév tov tetpddav npaypatikay apiOudy dote va petatpépa to R* oe daxttAro iaíocơn'Š H i6éa etvar va OewprHoovpe pta tetpdda (a1, A2,03,04) OTN HopgfÍ a,1 + dei + a3j + ask, va tpoobétoupe Katd tov auvAOn Todt

(a1 + aoi + a3j + ask) + (6:1 + boi + bạ] + bạk)=

— (a1 + b¡)1 T (a› + bo )i + (a3 + b3)j + (a4 + bạ)k

15 H tồéœ vœ Vewpet touc uryadixovc dovuoúÓc øœv ÉeUYđ@tđ Teayyatixay dœovuov oo0eÍÀetot Zz / , , z , , /

ơtov Hamilton ãou trị œvoœxớÀuÚec yUew oto 1830 O (Soc avapgoe: dtt thy e€hynoe otoUC HUXÐOÚC YtOÚC TOU ot oTolol xdV_E Tew! tov pwroboay «Agod UnauTé UTopeEtc xan ToAAATAdOLaCEIC ÉcUYyđotœ apiOuav umopeic va TOÀÀ@TtÀđŒOtLđOEtC XƠI TeLdSec;» xo AUTOS TOUc anavtoboE T&VTO

to (S10 «Oy Médvo va touc tpootéow xo va touc apaipgow Unoeo» H Wea tou opiouod

Z = Qa, — Gai — aaj — agk

Etons optlovpe to Hétpo z va etvai o tpaypatiKds api0uds

|z| = 4/0? + a3 + a2 + af

Kai pe KdmoIovs UToAoyIopovs prtopovpe va anodetEoupe tous Kavdves (via oTotd- dnmote x,y € H):

Aoxnon 46 Na detfete 6t1 Kd0e UTooMpa tov Tpaypatikay aptudy Ba rEepiéyel

TO OMA THY PNTAY

Aoxnorn 47 Na detéete 6t1 av p efvai tpatos aptHóc to ơứ/oÀo

{a+ b,/p: a,b € R}

efvai unoodmpa tou R

Aoxyor 48 Na deffete 6t1 n anerxdrion f;C > C pe ƒ(2) = 2, ónoU c Z oupBodtCovpe tov aucvyr pryadiKd tov z, efvai avtopopgiopids Xpnoiporoi@y tas

to Tapandve va arodetfete THY TaUTOTNTA:

(a? + b’)(c? + d’) = (ac — bd)? + (ad + be)?

Aoxnor 49 Na fpette to Kéytpo tov daxtvAiov diaipeons H twy tetpdday tou Hamilton

Aoxyor 50 Eotw M2(R) 0 daxttAws OAwy toy 2 x 2 mivdkwy tdva oto R Na ÔcÍÈ€t€ 6T1 TO UTOOUVVOAG ToU:

Z = Qa, — Gai — aaj — agk

Entons optlovpe to pétpo pitas tetpddas z va etvai o mpaypatixds apiOyuds

Aoxnon 53 Anodeiéte thy napakdtw tavtotnta tov Lagrange:

(aj + độ + gã + a2) (bị + bộ + 03 +07) = cf + G+ G+e4

Omou

qj = a,b; — 02s — 0aÖa — aby

Co = 010 + And, + a3b4 + agbs

Ca — đ1Öa — dob, + a3b, + abo C4 = a b4 + dob; — 0aba + a4b,

1.7 Ilenepacuevor AaxtvAtor - Medéty tov Z, -

Mepixéc epapuoyes otn Oewpia twv ApwWyuUwv

Ag ETIOTOEVOUUE OTHY TEPITTMON Twv TEeTEOPAOUEVUY SaxtUAiwy To Teato Tou Va ồcf6oUue Ít rị qdodX⁄ớt@ TOĨtŒOT;

IIeĩxơơn 12

Kdde renepacpévn axépaia nepioyn etvai odya

Anoddetgyn: Auté nou ĐếÀouuc vớ dSeifouue efvar dtr av Evac avtivetavetixdc dSaxtbAtog ye Uovdda A Eyer nenepacuevo TAHVos ototyeiwv xou Sev Eyer SiapetEc tou undevdc téte xaVe un undevixd ototyeio tou Va avtloteewETaL

Ác uroÙéøouue ĩtt youve Eva tétoto SaxtvAtlo R = {ap = 0, a; = 1, , an}

ue 2+ 1 otoryeta xa Eota a # 0 Eva onolodhxotE otoryeio tou SiapopEettxXd ANd TO

0 Tia va Seiouue Sti To a ŒvxtGtoé(0Etơt toếret vớ ồcÍÊOUUE OTtL UtđOXEL XớTOtO

r #0 ue ar = I ồnÀdồf ĩxt to Ì cĐớt øtouyeÍO toU ØUVĨÀOU

A={ar:reR\ {0}}

TIaoatnpouye to e€fc:

Av r,r’ civat 800 un undevixd otoryeta Stapopetiné UETAEY touc téte ar A ar’ nou

onuatver 6tt to A éyet œxotƯ6ưc tĩøœ ưtotyefÍœ ĩơơ xơt to R\ {0} Hodyyoti, av

ouveBave ar = ar” tĩte g(r — r") = Ú xơi cretồf cÍudơte Ø£ œ⁄Éo0Iđ TeOtOX4 1OÉREt

a=0'fr—r' =0 Tyouue 0xo0éòet ĩtt ø # Ú xơt XŒtdÀfyouue ott Va npéret r=r'

Extonc to 0 ¢ A, ya tov t8to Adyo Gna xu Tol

>uverœc 0œ éyouUe ĩtt 4 = F\ {0} xơ doale A O

Anĩ œurĩ EYOUUE oa TOPLOUAL TO TLOUPOAEATO:

IIeĩxơơn 13

O ơaKcúÀuoc 22„ cÍ⁄01 odpa ay Kai pidvo ay to n etvai mpa@tos apiOuds

Anoddeten: Av o Z,, civat omua tote dev Eye Stoupeétec tou undevdc, doa on Va

eivat Tea@tos (ytatt;)

Avttotpoga, av o n etvor me@toc Sev urnopel oto Zp va Eyouue Siaipétec tou

undevdc yrati av ya xdrora a,b © Zp \ {0} toyve ab = 0 téte Va etyoue nab oxĩte nia 4 nb, nov eíVœt ad0vato apod 1 < a,b < n—1 “uvenwd<, and tyv

ToorIyo0uevn noĩtdơn to 2 Va civar caua

Enetồf\ UxớoXOUV đ£tOOL T0G)TOL UTỚOXOUV X0 ỚTEIO% TETREOđOUÉVđ Ø@U0tO llœoœrrieefơre ĩtt ỒƯÚO TETEOđ%GUÉVGŒ Ø@Udt0 eÍVớt tGOO0(01/6 0V Xơt UĨVO ŒV ÉYOUV

to fồto tÀffoc ơtotyefev Aoyĩtecod Va efetd&oouue to ChtHUA av UmdeyoUYV đÀÀơ

xExeodouévd ơoudtd exvĩc Œrĩ tớ Z„ (xœt 9œ cÍ6ouuc ĩtt UYớoyOUV) X0t ETÍOTC

AV UTđOXOUV TETEOđ%GUẺVŒ% Ø@udtd urn) œvxrtHetdœDctt⁄ớ H andvtjon oe autd to Eodtriud cÍơt dovrittxf, mĩc ồeÍyVet to Tapaxdtw oNUAvTLXd VEewonua.

Ocaperyua 9 (Wedderburn)

Kade renepacpévo odpa etvai avtipetatetixd

H anddegn tou Vewortuatoc autov elvar oyetixa SUoKoAH “or Va yiver apou ồ@- ØOUUE XdTOLES AXON Evvoles THC Vewplac SaxtuAtwv

Ác cmtơto€oUUe ØtoUC TEtEoŒ0UÉVoUC SaxTUALOUC THC WopOTS Zp xa ac Ete- TAGOUUE TOTE Eva GtOLyElo TOU Elvat N AVTLOTEEUILO Wo TEOC TOV TOAAATAGCLAGUO

H exduevyn nodtaon dtvet ura TAVEN ANdvTNON GE AUTO TO EOWTHUA XO TOOPAVOS øuverớyetœt trì lloótơơn 13

IIeóxơzơn 14

Eva pn pnoevixd atotyeto a tov Z,, elvai avtiotpéynpio ay Kai n p<H H pdvo ay o1a,n ; etvai ơx€tiKá tp@tot apiOyot, dnAadh évouy cay péyioto Kod diaipétyn tv povdda

Anoddetgn: Ac unodéoouue Ott ol a, n Elva OyETIXk TOETOL xaL OUVETaS Da UTdE-

youv dbo axéeator x,y ue 1 = ax + yn, dydad4 1 = ax (mod n) Av Dewohoouue

£1 € Z„ Ue #¡ = # (mod n),téte BAExouuE Ott az; = 1 (mod n) xoU öriudfVet ótt

o a €xet avtioteo@o oto Z, Tov 2}

Avtiotpowa, av o a avtlotpég~etat Tote yia xdnoto x EC Z, Ya Eyouue dt (oto Zn) ax = 1 4 toodbvaua ax = 1 (mod n) fh toodbvaua Ya undeyer y € Z ve

, , f !, f + +

ax+ny = 1 Add& téte o1 a,n Va nOoÉRet va Elvan TE@TOL UETAEU TOUg aAPOU O

UeYtGtoc xowóc ồtœtoérnc 0ơ Stapovoe to 1 Kou cuvenas Va ouvemimte ue aut OO

Tia x&0e @uơt⁄ó dœotÔUó ac opioouue O(n) va efivat to TÀfÙOC T6V (0UƠIX@V TOU

EÍVƠL UIXOÓtEOOL ATO N XXL TOWTOL WC TEOG N, SNAASH to TAHVOS Twv OTOLyEiwy TOU

ouvdhou {m EN:1<m<n—1 xm (m,n) = 1} Opileton Etor uia øuvớotrion)'8

@:N—-N 7 onola Aéyeta 7 ouvéetyjon tou Kuler xo natCer onuavttxd eddo ơtr Vewpia tov aorduayv Mnopovue va utoAoyioouue thy tly O(n) av yywoiTouue THY avaAUoN TOU N OE YIVOUEVO TO@TwWY AUTO TEOXUTTEL ANd THY «KOAAATAACLAOTIXY tỒtÓt7it%> THC GUVdETHONS TOU onatver 6tt av S40 Quolxol mM, n Elva TE@TOL UETAEY

toug tote d(mn) = o(m)d(n) Enedh ot Evvoirec autéc ralfouv onavttxd póÀo

ỒÍVoUue Evav exionuo OPO Yt ŒUtéc

Ocuouóec 30

Mía aptfr†rt£Kr1† ovrvdptynon ctvai pita ovvdptnon f : NC

Mia apitunuky ovvdptnon f 0a Aéyetai noAAarAaotaotiKH ay yia oroiadrrote m,néN nov etvar rodtot petags tous ioxvei

f(mn) = f(m)f(n)

18H ouvdetyon éyet neồío optouo0 touc un undsewxole Quoixobs aAA& Uxooo0ue ơuudrtxớ

va éơouuec (0) = 0