PDF Abstract algebra and famous impossibilities: squaring the circle, doubling the cube, trisecting
Abstract Algebra and Famous Impossibilities: Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations 2nd Edition Sidney A. Morris
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Elementary abstract algebra: examples and applications
That p andearetranscendentalnumberswasprovedinthe firstedition.Inthis editionexcitingadvancesintranscendentalnumbertheorywhichhaveoccurredin the130yearssinceLindemannprovedthat p istranscendentalareincluded.Ithank TabokaPrinceChalebgwaforadviceonthesectiononTranscendentalNumber Theory.
Asstudentswereverycomplimentaryaboutthegentlestyleofpresentationin the firstedition,everyattempthasbeenmadetomaintainthatinthisedition. Anotherfeatureofthiseditionistheinclusionofmuchmorehistoricalinformation. Thebibliographyhasalsobeengreatlyexpanded.
The firsteditiongrewoutofasecond-yearsubjectwhichArthurJones,Ken Pearson,andIhadtaughtformanyyearsatLaTrobeUniversityinMelbourne, Australia.Thissecondeditionisaimedatsimilarstudents.Butitshouldalsobe mentionedthattheknowledgeinthisbookwouldbebeneficialtofuture(and
ProblemIisthatofconstructingwithcompassandstraightedgeacubehaving twicethevolumeofagivencube.Ifthesideofthegivencubehaslength1unit, thenthevolumeofthegivencubeis13 ¼ 1.Sothevolumeofthelargercube shouldbe2,anditssidesshouldthushavelength 2 3 p . Hencetheproblemis reducedtothatofconstructing,fromasegmentoflength1,asegmentoflength 2 3 p .
ProblemIIistoproduceaconstructionfortrisectinganygivenangle.Whileitis easytogiveexamplesofparticularangleswhichcanbetrisected,theproblemisto giveaconstructionwhichwillworkfor every angle.
Arealnumber c issaidtobe constructible if,startingfromalinesegmentof length1,wecanconstructalinesegmentoflength jcj ina fi nitenumberofsteps usingstraightedgeandcompass.
Weshallprove,inChapters 5 and 6,that arealnumberisconstructibleifand onlyifitcanbeobtainedfromthenumber1bysuccessiveapplicationsofthe operationsofaddition,subtraction,multiplication,division,andtakingsquare roots.Thus,forexample,thenumber
isconstructible.
Now 2 3 p doesnotappeartohavethisform.Appearancescanbedeceiving, however.Howcanwebesure?Theanswerturnsouttobethatif 2 3 p didhavethis form,thenacertainvectorspacewouldhavethewrongdimension!Thissettles ProblemI.
addition,subtraction,multiplication,anddivision(exceptby0) canbeperformedwithoutrestriction.Thesameistrueoftheset R ofallrealnumbers, whichisanotherexampleofafield.Likewisetheset C ofallcomplexnumbers,with theaboveoperations,isafield.
Aformaldefinitionofafieldisgiveninthe AppendixtoChapter1.Strictly speaking,afieldconsistsofaset F togetherwiththeoperationstobeperformedon F.
Whenthereisnoambiguityastowhichoperationsareintended,wesimplyreferto theset F asthefield.
If F isafieldand E ⊆ F itmayhappenthat E alsobecomesafieldwhenweapply theoperationson F toitselements.Ifthishappenswesaythat E isa subfield of F and,reciprocally,that F isan extensionfield of E.Thissituationisoftendenotedby F/E andreadas“F over E”.Asparticularcases, Q isasubfieldofboth C and R while R and C arebothextensionfieldsof Q.Wewritethisas C/Q and R/Q
Animportantrelationshipbetween Q and C isexpressedinthefollowingproposition.
Thereforeeachpositiveintegerisin F,as F isclosedunderaddition.
Soeachnegativeintegerisin F,as F isclosedundersubtraction.
Also 0 ∈ F,as F isafield.
Hencetheintegers r and s arein F
Since F isclosedunderdivision(and s = 0),thequotient r /s ∈ F
So x ∈ F ,asrequired.
Thereare,infact,lotsofinterestingfieldswhichliebetween Q and C andwe shalldevotemuchtimetostudyingthem.
Ofcoursenotallfieldsaresubfieldsof C.Toseethis,considerthefollowing.For eachpositiveinteger n put
Zn ={0, 1, 2,..., n 1}.
Wedefineadditionofelements a and b in Zn by
a ⊕n b = a + b (mod n );
thatis,add a and b intheusualwayandthensubtractmultiplesof n untiltheanswer liesintheset Zn .Wedefinemultiplicationsimilarly: a ⊗n b = a × b (mod n ).
Forexample,
3 ⊕6 4 = 1 (subtract6from7),
2 ⊕6 1 = 3,
3 ⊗6 4 = 0 (subtract12from12),
9 ⊗10 7 = 3
Itcanbeshownthatif n isaprimenumberthen Zn isafield.Itisobviousthat inthiscase Zn isnotasubfieldof C.Forexample,if n = 2, 1 ⊕2 1 = 0 in Z2 ,but 1 + 1 = 2 in C;sotheoperationofadditioninthefield Z2 isdifferentfromthatin thefield C.Unlikeourpreviousexamples Q, R and C,thefield Zn isa finitefield ; thatis,ithasonlyafinitenumberofelements.
Thustheset Z ofalltheintegersisanexampleofaringwhichisnotafield, whereastheset N ofallnaturalnumbersisnotaring.
Weevenallowtheconceptofmultiplicationitselftobeliberalized.Wedonot requireittobecommutative;thatis, ab maybeunequalto ba .Wealsopermitzero divisors,sothatitispossibletohave
ab = 0 butneitheranorbiszero
Observethatif M isthesetofall 2 × 2 matriceswithentriesfrom R,then M isaringwiththeringoperationsbeingmatrixadditionandmatrixmultiplication. However, M isnotafieldsinceif
A = 12 34 and B = 56 78
then AB = 1922 4350 = BA = 2334 3146
Notealsothatthisring M haszerodivisors;forexample,if
Recallfromlinearalgebrathata vectorspace consistsofaset V togetherwithafield F.Theelementsof V arecalled vectors andtheelementsof F arecalled scalars Anytwovectorscanbeaddedtogiveavectorandavectorcanbemultipliedbya scalartogiveavector.Aformaldefinitionofavectorspaceisgiveninthe Appendix toChapter1
Whenthereisnodangerofambiguityitiscustomarytoomitreferencetoboth thescalarsandtheoperationsandtoregard V itselfasthevectorspace.Inourstudy, however,thefieldofscalarswillbeconstantlychangingandtoomitreferencetoit wouldalmostcertainlyleadtoambiguity.Henceweshallincludethefield F inour descriptionandreferto“thevectorspaceVover F”.
If v1 ,v2 ,...,vn arevectorsinthevectorspace V overthefield F,thenwesay thattheset {v1 ,v2 ,...,vn } ofthesevectors spans V over F providingthatevery vector v ∈ V canbewrittenasalinearcombinationof v1 ,v2 ,...,vn ;thatis, v = λ1 v1 + λ2 v2 + ... + λn vn
Theset {v1 ,v2 ,...,vn } issaidtobea basis for V over F ifitislinearlyindependent over F andspans V over F.
Whileafinite-dimensionalvectorspacecanhaveaninfinitenumberofdistinct bases,thenumberofvectorsineachofthebasesisthesameandiscalledthe dimension ofthevectorspace.Recallalsothat,inavectorspaceofdimension n ,anyset withmorethan n vectorsmustbelinearlydependent.(Forexample,anysetoffour vectorsinathree-dimensionalspaceislinearlydependent.)
Themostintuitiveexample,inthisregard,isthevectorspace C over R inwhich wetake C forthevectorsand R forthescalars.Thisvectorspacearisesfromthe pairoffields C and R,inwhich R isasubfieldof C.Wemaythinkoftheelements of C aspointsintheplaneandthenthevectorspaceoperationscorrespondexactly totheusualvectoradditionandscalarmultiplicationintheplane.
Moregenerally,if E and F arefieldswith E asubfieldof F,wemaytake F forthe vectorsand E forthescalarstogetthe vectorspace F over E.(Notethatthesmaller fieldisthescalarsandthelargeroneisthevectors.)
1.1.2Definition. Ifthevectorspace F over E isfinite-dimensionalitsdimensionis denotedby [F : E] andcalledthe degreeof F over E
Forexample,itisreadilyseenthat [C : R]= 2 and [R : R]= 1.
(a)Provethat F isavectorspaceover Q andwritedownabasisforit.
(b)∗ Provethat F isafield.
(c)Findthevalueof [F : Q]
6.Aring R issaidtobean integraldomain ifitsmultiplicationiscommutative,if thereisanelement1suchthat 1 x = x 1 = x forall x ∈ R ,andifthereareno zerodivisors.
(a)Is Z4 anintegraldomain?
(b)∗ Forwhat n ∈ N is Zn anintegraldomain?
7.Let E beasubfieldofafinitefield F suchthat [F : E]= n ,forsome n ∈ N.If E has m elements,forsome m ∈ N,howmanyelementsdoes F have?(Justify youranswer.)
8. (a)If A , B ,and C arefinitefieldswith A asubfieldof B and B asubfieldof C , provethat [C : A ]=[C : B ].[ B : A ]. [Hint.UsetheresultofExercise7.]
(b)Iffurthermore [C : A ] isaprimenumber,deducethat B = C or B = A .
Thecoefficients a0 , a1 ,..., an areassumedtobelongtosomering R .Theabove expressionisthencalleda polynomialoverthering R . Asyetwehavesaidnothingaboutthesymbol x appearingintheaboveexpression.Thereare,infact,twodistinctwaysofinterpretingthissymbol.Oneofthese waysleadstotheconceptofa polynomialform,theothertothatofa polynomial function.
PolynomialForms
Toarriveattheconceptofapolynomialform,wesayaslittleaspossibleaboutthe symbol“ x ”.Weregard x anditsvariouspowersassimplyperformingtherôleof “markers”toindicatethepositionofthevariouscoefficientsintheexpression. Thus,forexample,weknowthatthetwopolynomials
1 + 2 x + 3 x 2 and2 x + 3 x 2 + 1
1.2Polynomials7
areequalbecauseineachexpressionthecoefficientsofthesamepowersof x are equal.
wemayregardthepowersof x assimplytellinguswhichcoefficientstoaddtogether. Theyplayasimilar,althoughmorecomplicated,rôlewhenwemultiplytwopolynomials.
Although x isregardedasobeyingthesamealgebraicrulesastheelementsofthe ring R ,wedonotthinkofitasassumingvaluesfrom R .Forthisreasonitiscalled an indeterminate.
Foremphasis,weshalluseuppercaselettersforindeterminatesinthisbookand write“ X ”insteadof“ x ”soourpolynomial(1)becomes
andwecallita polynomialform (overthering R ,intheindeterminate X ).
Whatreallymattersinapolynomialformarethecoefficients.Accordinglywe saythat twopolynomialformsareequalifandonlyiftheyhavethesamecoefficients. 1.2.1Definitions. Ifintheabovepolynomialformthelastcoefficient an = 0,then wesaythatthepolynomialhas degree n .
Whenwedonotwishtostatethecoefficientsexplicitlyweshallusesymbolslike f ( X ), g ( X ), h ( X ) todenotepolynomialsin X .Thedegreeofanonzeropolynomial f ( X ) isdenotedby deg f ( X ).
Ifintheabovepolynomialform an = 0,wecall an the leadingcoefficient. Thecollectionofallpolynomialsoverthering R intheindeterminate X willbe denotedby R [ X ]
Inparticular,ifthering R happenstobeafield F,weobtainthepolynomialring F[ X ].Thisringwillnotbeafield,however,since X ∈ F[ X ] but X hasnoreciprocal in F[ X ] because X f ( X ) = 1, forall f ( X ) ∈ F[ X ]
PolynomialFunctions
Backintheoriginalexpression(1),analternativewaytoregardthesymbol“ x ”isas avariablestandingforatypicalelementofthering R .Thustheexpression(1)may beusedtoassigntoeachelement x ∈ R anotherelementin R .Inthiswaywegeta function f : R → R withvaluesassignedbytheformula
Suchafunction f iscalleda polynomialfunction onthering R
Thusifweregardpolynomialsasfunctions,emphasisshiftsfromthecoefficients tothevaluesofthefunction.Inparticular, equalityoftwopolynomialfunctions f : R → R and g : R → R meansthat
f ( x ) = g ( x ), forall x ∈ R , whichisjustthestandarddefinitionofequalityforfunctions.
Clearlyeachpolynomialform f ( X ) determinesauniquepolynomialfunction f : R → R (becausewecanreadoffthecoefficientsfrom f ( X ) andusethemto generatethevaluesof f viatheformula(2)).Istheconversetrue? Doeseachpolynomialfunction f : R → R determineauniquepolynomialform? Theansweris: notnecessarily! Thefollowingexampleshowswhy.
1.2.2Example. Let f ( X ) and g ( X ) bethepolynomialformsoverthering Z4 given by
Proof. Thesepolynomialformsdeterminetwofunctions f : Z4 → Z4 and g : Z4 → Z4 ,withvaluesgivenby
f ( x ) = 2 x and g ( x ) = 2 x 2 .
Hencecalculationin Z4 showsthat
f (0) = 0 = g (0)
f (1) = 2 = g (1)
f (2) = 0 = g (2)
f (3) = 2 = g (3).
So f ( x ) = g ( x ),forall x ∈ Z4 .Thusthefunctions f and g areequal. Thisissomethingofanembarrassment!Wehaveproducedanexampleofaring R andtwopolynomialforms f ( X ), g ( X ) ∈ R [ X ] suchthat
willbemoreorlessthesamewhile f ( x ) willbequitedifferent,beingthevalueof f at x .
Exercises 1.2
1.Giveanexampleofanelementofthepolynomialring R[ X ] whichisnota memberof Q[ X ].
2.Ineachofthefollowingcases,giveapairofnonzeropolynomials f ( X ), g ( X ) ∈ Q[ X ] whichsatisfiesthecondition:
(i) deg( f ( X ) + g ( X ))< deg f ( X ) + deg g ( X );
(ii) deg( f ( X ) + g ( X )) = deg f ( X ) + deg g ( X ); (iii) deg( f ( X ) + g ( X )) isundefined.
3.(a)Let F beafield.Verifythatif f ( X ) and g ( X ) arenonzeropolynomialsin F[ X ] then f ( X ) g ( X ) = 0 and
deg f ( X ) g ( X ) = deg f ( X ) + deg g ( X ).
(b)Deducethatthering F[ X ] isanintegraldomain. (See Exercises1.1#6 forthedefinitionofintegraldomain.)
(c)Is Z6 [ X ] anintegraldomain?(Justifyyouranswer.)
4.Let f ( X ) denotethepolynomialformoverthering Z6 givenby f ( X ) = X 3 . Findapolynomialform g ( X ),with f ( X ) = g ( X ),suchthat f ( X ) and g ( X ) determinethesamepolynomialfunction.
5.∗ Let R beanyfinitering.Provethatthereexistunequalpolynomialforms f ( X ) and g ( X ) over R suchthat f ( X ) and g ( X ) determinethesamepolynomial function.
