Introduction to Abstract Algebra : Rings and Fields

21 Pages·2013·0.188 MB·English

by  Rachel Quinlan

MA314 (Part 2) 2012-2013 : Lecture Notes
Introduction to Abstract Algebra : Rings and Fields
DrRachelQuinlan
SchoolofMathematics,StatisticsandAppliedMathematics,NUIGalway
March22,2013
Contents
IntroductionandCourseInformation 2
1 RingsandFields-Examples 4
2 AxiomaticDefinitions 7
3 PolynomialsandRoots 10
4 TheEuclideanAlgorithm 14
5 ThefiniteringsZ/nZ 18
6 Morefinitefields-extensionsofZ/pZ 20
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MA314(PART2)2012-13: RINGSANDFIELDS
LECTURER DrRachelQuinlan
RoomC105,GroundFloorA´rasdeBru´n
[phone](Extension3796frominsideNUIGalway)
[email protected]
http://www.maths.nuigalway.ie/∼rquinlan
StudentsarewelcomeinmyofficewheneverIamthere
LECTURES Tuesday1.00LarmorTheatre,Friday12.00AM150
SYLLABUS
Thisisashort(10lectures)introductiontothealgebraictheoryofringsandfields,whicharecom-
ponentsofthewidersubjectofabstractalgebra. Thephilosophyofthissubjectisthatwefocuson
similaritiesinarithmeticstructurebetweensets(ofnumbers,matrices,functionsorpolynomials
forexample)whichmightlookinitiallyquitedifferentbutareconnectedbythepropertyofbeing
equippedwithoperationsofadditionandmultiplication. Thesetofintegersandthesetof2by
2 matrices with real numbers as entries are examples of rings. These sets are obviously not the
same,buttheyhavesomesimilarities-andsomedifferences-intermsoftheiralgebraicstructure.
Althoughpeoplehavebeenstudyingspecificexamplesofringsforthousandsofyears,theemer-
gence of ring theory as a branch of mathematics in its own right is a very recent development.
Muchoftheactivitythatledtothemodernformulationofringtheorytookplaceinthefirsthalf
ofthe20thcentury. Ringtheoryispowerfulintermsofitsscopeandgenerality,butitcanbesim-
ply described as the study of systems in which addition and multiplication are possible. Fields
arespecialexamplesofrings,inwhichmultiplicationiscommutativeandinwhichitispossible
to divide by any non-zero element. Examples of fields include the field Q of rational numbers
andthefieldCofcomplexnumbers.
Thelecturenotescontainalistofmorespecifictopics,itemizedoverthetenlectures.
LEARNINGOUTCOMES
Bytheendofthiscourseyouwillbeableto:
• Explain the meaning of the terms ring and field and provide and identify examples (and
non-examples)ofringsandfields.
• Discusstheimportantfeaturesofgivenexamplesofringsandfields.
• ImplementtheEuclideanalgorithm.
• Constructexamplesoffiniteringsandfieldsandperformcalculationsinthem.
ASSESSMENT
End-of-SemesterExaminations:Onetwo-hourexaminationinMA314.Morelaterontheformatand
onwhattoexpect.
ContinuousAssessment: Therewillbetwowrittenassignmentsoverthecourseofthe10lectures.
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RESOURCES
OnlineresourcesforthiscoursewillbemaintainedontheMA314Blackboardpage(clickon“Part
2 : Rings and Fields” in the main menu). These will include lecture notes which constitute the
“text”forthecourse. Youareexpectedtostudythelecturenotes, whicharemoredetailedthan
the discussion in lectures. As well as the “official” assignments, there are exercises throughout
thelecturenotes. Youareadvisedtothinkabouttheseandtomaintainyourownexpandedsetof
notesincludingyourresponsestotheseexercises.
Forsupplementaryreading,youmayfindthefollowingbookshelpful:
• AFirstCourseinAbstractAlgebra,JohnB.Fraleigh(512.02)
• ModernAlgebra,JohnR.Durbin(512.02DUR)
AWORDOFADVICE
Atthisadvancedlevel, Mathematicsismoreaboutunderstandingandexplainingconceptsand
theirlogicalconnectionsthanaboutdoingcalculuationsor“workingoutanswers”. Thismeans
that it involves learning a formal and technical language, which takes some practice. Don’t be
discouragedifittakesyousometimetogetusedtothisinthecontextofabstractalgebra-having
reachedthispointyouarecapableofit.
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1 Rings and Fields - Examples
Webeginwithalistoffamiliaralgebraicstructuresthatwillbeusefulforreferenceandasasource
ofexamplesasweproceed.Alloftheseobjectsareprobablywellknowntoreadersofthesenotes,
butitisimportanttomakesurethatyouarealsofamiliarwithallthenotationusedhereandthat
youcanusethisnotationinapreciseandcarefulway,bothinreadingandinwriting.
NOTE For these lectures you will need to be familiar with the notations N,Z,Q,R,C and their
meanings. Youwillneedtoknowhowthesenumbersystemsrelatetoanddifferfromeachother
andbeabletorecallthisinformationeasily. Ifyouarerustyonthisatall,havealookforexample
attheWikipediapageon“Number”-en.wikipedia.org/wiki/Number
1. Thesetofintegers,Z.
Z={...,−2,−1,0,1,2,3,...}
TheoriginofthesymbolZcomesfromZahlen,theGermanwordfornumbers.
2. Thesetofrationalnumbers,Q.
(cid:10) (cid:11)
a
Q= :a∈Z, b∈Z,b(cid:54)=0
b
Q is the set of all numbers that can be expressed as the quotient of two integers, in other
wordsasafractionwithintegersasnumeratoranddenominator(thedenominatormustbe
non-zero obviously). The notation Q comes from “quotient”. Note that Z ⊂ Q (i.e. every
integerisarationalnumber;ZisasubsetofQ).
3. Theset2Zofevenintegers.
2Z={2k:k∈Z}={...,−4,−2,0,2,4,6,...}
4. ThesetM (R)ofreal3×3matrices.
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Elements of this set are 3×3 matrices whose entries are real numbers (R is the set of real
numbers-therealnumbersincludeallpointsonthenumberline). Notethatifwecanadd
or multiply any two of these 3×3 matrices. In each case the result will again be a 3×3
matrixwithrealentries.
5. Q[x]-thesetofpolynomialsinxoverQ
ApolynomialinxoverQisanexpressionoftheform
adxd+ad−1xd−1+···+a1x+a0,
wheredisanon-negativeinteger, anda0,a1,...,ad arerationalnumbers(elementsofQ).
So a polynomial (in x over Q) involves a constant term and a finite number of positive
integerpowersofxwhichappearwithrationalcoefficients. Thefollowingareallexamples
ofelementsofQ[x]:
1 21
x+2, x2−5, x4+ x3+x, , 0
2 5
NotethatrationalnumbersthemselvesareincludedinQ[x],asthoseelementsinwhichthe
positivepowersofxallhavezerocoefficient.
WecandefineZ[x],R[x]etc. inthesameway-theQ,Z,Rorwhateverindicateswherethe
coefficientslive.
6. D (Z)-thesetofdiagonal2×2matriceswithintegerentries.
2
(cid:12) (cid:13)
(cid:18) (cid:19)
a 0
D (Z)= :a,b∈Z .
2 0 b
Note that the sum of two diagonal matrices is diagonal, and the product of two diagonal
matricesisdiagonal.
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7. UT (R)-thesetofuppertriangular3×3matriceswithrealentries.
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 
 a b c  
UT3(R)= 0 d e :a,b,c,d,e,f∈R.
0 0 f
Again, verify that the sum and product of two upper triangular matrices are again upper
triangular.
8. C-thesetofcomplexnumbers.
C={a+bi:a,b∈R, i2 =−1}
Thesetofcomplexnumbersisobtainedfromthesetofrealnumbersbyadjoiningan“imag-
inary”squarerootof−1,denotedbyi. Complexnumberscanbeaddedtogetherandmul-
tipliedtoproducenewcomplexnumbers.
9. Q(i)-thesetofGaussianrationalnumbers
Q(i)isthesubsetofCconsistingofallthosenumbersa+biwhoserealpartaandimaginary
√
partbarebothrational. Soforexample1+3ibelongstoQ(i)but1− 2idoesnot. Note
thatQ(i)isasubsetofCbutnotofR.
EXERCISE: Show that the sum and product of two Gaussian rational numbers is always
a Gaussian rational number. (This is not completely obvious at first glance - not every
complex number belongs to Q(i), so you have to explain why taking the sum or product of
twoelementsofQ(i)couldn’ttakeyououtsideQ(i)).
Thesesetsarealldifferent,andtheirelementsaremathematicalobjectsofdifferentkinds(in-
tegers, realnumbers, complexnumbers, matrices, polynomialsetc.). However, intermsoftheir
algebraic structure, they have some overall similarties. All of them are naturally equipped with
operationsofaddition,subtractionandmultiplication. Thismeans
givenanypairofelementsofanyofthesesets,itispossibletoaddthemtogether,multiplythem,or
subtractonefromtheother,TOGETANOTHERELEMENTOFTHESAMESET
Thisstructuralsimilaritybetweenallthesesets(andmanyothers)isessentiallythethemeofthe
mathematicaltheoryofrings. Afewpointstonote:
1. Itisimportanttonotethatwhenwetalkaboutthesesetsbeing“equipped”withaddition,
multiplication and subtraction, we mean not only that it makes sense to add, subtract or
multiply any two elements of the set, but that when we do so the result is still within the
same set (we could express this by saying that each of our sets is closed under addition,
multiplicationandsubtraction).
In this sense the set N of natural numbers (or positive integers) is not equipped with a
subtraction operation. While it is true that it is always possible to subtract one natural
number from another, doing this might take us outside the set of natural numbers. For
example5and8arebothnaturalnumbersbutthedifference5−8=−3isnot. SoNisNOT
closedundersubtraction.
Similarly,ifweweretoconsidertheset{...,−3,−1,1,3,5,...}ofoddintegersinthiscontext,
itwouldnotbeclosedunderaddition(orsubtraction),sincethesumoftwooddintegersis
notanoddinteger.Ontheotherhand,thesum,differenceandproductoftwoevenintegers
isalwaysaneveninteger,sothesetofevenintegerscanmakeitintoourlist.
Itisnotautomaticallyobviousthatallofthesetsinourlistareclosedundertheirownad-
dition,subtractionandmultiplicationoperations. Forexampleitisnotcompletelyobvious
that the product of two upper triangular 3×3 matrices is again upper triangular. These
thingsneedtobecheckedandyouareadvisedtocheckthem.
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2. The words “addition” and “multiplication” are being used here in a generic way. For ex-
amplemultiplicationof3×matricesisnotthesamethingasmultiplicationofrealnumbers
orofpolynomials, butthesameword“multiplcation”isusedforallofthese-thespecific
mechanismofwhat“addition”or“multiplication”involvesdependsonthecontext.
Informal and Incomplete Definition: A ring is an algebraic structure in which it is possible to add,
multiplyandsubtractelementsasintheexamplesabove.
Thereasonwhythisdefinitionisincompleteisbecauseitdoesnotspecifywhatpropertiesthe
additionandmultiplication(andsubtraction)operationsshouldhaveorhowtheyshouldinteract
witheachother.
Themultiplicationoperationsinourexamplesdonotallbehaveinthesameway. Thereare
twofeaturesofmultiplicationthatareofinterestrightnow.
• First,multiplicationmayormaynotbecommutative.Thismeansthattheorderinwhichtwo
elementsaremultipliedmightormightnotmakeadifference. Forexample,multiplication
ofintegersiscommutative,becauseab=bawheneveraandbareintegers. Multiplication
of3×3matricesisnotcommutative,becauseABmaynotbeequaltoBAfor3×3matrices
AandB.
EXERCISE: Determine in which of our examples the multiplication is commutative. (The
hardestexampleoftheseisprobablyUT (R).)
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• Insome(but)ofourexamples, itispossibletodividebyanynon-zeroelements(andstay
withintheset). ThisisnotthecaseintheringZofintegers,becausewhilewemaydivide
oneintegerbyanother,theresultofthis(e.g. 3)mighttakeusoutsideZ. WecandivideinQ
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byanynon-zeroelement,basicallybecausethereciprocalofanon-zerorationalnumberis
anothernon-zerorationalnumber. ThesameistrueinthesetofcomplexnumbersC-recall
thatifz=a+biisa(non-zero)complexnumber,then
1 1 1 a−bi a−bi a b
= = × = = − i.
z a+bi a+bi a−bi a2+b2 a2+b2 a2+b2
Sothereciprocalofanon-zerocomplexnumberisanon-zerocomplexnumber,andinthe
ringofcomplexnumberswehaveadivisionoperation(thesemeansthatwecandivideby
anynon-zerocomplexnumber).
We do not have a division operation in the polynomial ring Q[x] - i.e. it is not in general
possibletodivideonepolynomialbyanotherandgetapolynomialastheresult.
We do not have a notion of division for matrices either (although some matrices have in-
versesandsomedonot).
EXERCISE: Showthatthereciprocalofeverynon-zeroelementoftheringQ(i)isalsoanelement
ofQ(i).
Thismeans: ifaandbarerationalnumbers,showthattherealandimaginarypartsof 1 are
a+bi
alsorationalnumbers.
Informal Definition: A ring in which the multiplication is commutative and in which it is possible to
dividebyanynon-zeroelement(andstayinsidethering)iscalledafield.
EXERCISE: Whichofourexamplesarefields?
To answer this : you can rule out any example in which the multiplication is not commutative
straight away. For the commutative examples, what you need to do is figure out whether every
non-zeroelementhasareciprocalthatbelongstothering.
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2 Axiomatic Definitions
This section contains the “official and complete” definitions of the terms ring and field. These
consistofaxiomsspecifyingthepropertiesthatthedivisionandmultiplicationoperationsmust
haveandhowtheyinteractwitheachother.
Thesedefinitionsmakenoreferencetoanyparticularsetortothenatureoftheobjectsthatare
elementsofthering-whethertheyarenumbers,functions,matricesorwhatever. Whenthinking
about the definitions, try to bear this in mind and try also to bear in mind that “addition” and
“multiplication”arejustwordsdescribingtwowaysofputtingpairsofelementstogethertoget
newelementsintheset. Itcan’tbeassumedthattheyalwayscorrespondtofamiliarversionsof
multiplication and addition - that is why in the definition these operations are characterized by
their properties. This in a way is the whole point of abstract algebra - to characterize algebraic
structuresintermsoftheirpropertiesratherthanintermsofthedetailsofspecificexamples.
Definition(Formaldefinitionofaring)AringRisanon-emptysetequippedwithoperationsof
addition(+)andmultiplication(×)satisfyingthefollowingaxioms.
(ThefirstfiveaxiomsareconcernedwithadditionandarelabelledA1toA5forthisreason.)
A1 Risclosedunderaddition. Thismeansthatr+s∈Rwheneverr∈Rands∈R.
A2 Theadditionoperationiscommutative. Thismeansthatr+s=s+rforallpairsofelements
rands.
A3 Theadditionoperationisassociative. Thismeansthat(r+s)+t=r+(s+t)forallelements
r,sandtofR.
A4 R contains an identity element or neutral element 0R for addition (this is often referred to as
thezeroelementbecauseforadditionitresemblesthenumber0). Thepropertythatthezero
elementmustsatisfyis
r+0R =r(=)R+r)forallr∈R.
Soaddingthezeroelementtoanotherelementhasnoeffectonthatelement(hencetheterm
“neutral”although“identity”ismoreusual).
A5 ForeveryelementrofR,thereexistsanelement−rinRforwhich
r+(−r)=0R(=(−r)+r).
So−risthe“negative”or“additiveinverse”ofr.
NOTE: Itistheexistenceofadditiveinversesthatmakesitpossibletodefineasubtractionopera-
tion-subtractingaparticularelementfromsomethingmeansaddingthenegativeofthatelement.
Foraset(withadditionandmultiplication)tobeconsideredaring,itsadditionmustsatisfy
allofaxiomsA1toA5. ForexamplethesetofoddintegersisnotaringbecauseitfailsaxiomA1
(andA4). Theset{0,1,2,3,...}ofnon-negativeintegersisnotaringbecauseitfailsaxiomA5.
Forthemultiplicationoperation,onlyonespecialproperty(apartfromclosure)isspecified.
M1 IfrandsareelementsofR,thenr×sands×rarealsoelementsofR.
M2 ThemultiplicationinRisassociative. Thismeansthatforallelementsr,s,tofRwehave
(r×s)×t=r×(s×t).
AxiomM2meansthatifwewanttomultiplyelementsr,sandt(inthatorder),wedon’thaveto
specifywhetherwefirsttaker×sandthenmultiplythatontherightbyt, orwhetherwefirst
takes×tandthenmultiplythatontheleftbyr.
Thelasttwoaxiomsspecifyhowtheadditionandmultiplicationoperationsinteract-aswemight
expectwehavedistributivityofmultiplicationoveraddition. Thismeans:
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D1 Foranyelementsr,sandtofR,
r×(s+t)=(r×s)+(r×t)
D2 Foranyelementsr,sandtofR,
(r+s)×t=(r×t)+(s×t)
Soaringisanon-emptysetequippedwithadditionandmultiplicationoperationsthatsatisfyallofthese
axioms.
REMARK: Composingthislistofaxiomsmayseematfirstglancelikealotofunnecessaryfussing
todoaboutsystemsofarithmeticwithwhichweareallveryfamiliar. Ifyouarepuzzledabout
therationalebehindthis,onethingtobearinmindisthatthepointisnotjusttogiveanabstract,
formaldescriptionofthingsthatarefamiliar,butalsotocharacterizeadditionandmultiplication
bytheirabstractpropertiessothatourdefinition(andwhatevertheorymightbedevelopedfrom
it)appliesalsotosituationswheretheadditionandmultiplicationworkinlessfamiliarandless
obviously“natural”ways.
Another way to think about it : if someone asked you what “addition” means, what would
yousay? Itishardtoimaginetryingtoexplainwhatadditioniswithouthavingtofirststartwith
somethingsthatcanbeaddedtogether,likeintegersforexample. ThedescriptioninAxiomsA1
toA5abovegivesusaversatilewayofdescribingtheconceptofanadditionoperation(forrings)
thatdoesnotrelyonanyparticularcollectionofthingstobeaddedtogether.
Now that we know what a ring is, we turn our attention to fields. A field is a ring whose
multiplicationsatisfiesthreeextraconditions.
Definiton(Formaldefinitionofafield).
AfieldFisaringwhosemultiplicationoperationsatisfiesthefollowingadditionalproperties(the
label“FM”standsfor“fieldmultiplication”).
FM1 Themultiplicationiscommutative,i.e.
x×y=y×xforallx,y∈F.
FM2 Thereisanelement1F(differentfromthezeroelement)whichisanidentityelementorneutral
element with respect to multiplication. Multiplication by this element has no effect. This
meansthatforallelementsxofF
x×1F =x(=1F×x).
FM3 Ifxisanon-zeroelementofF(thismeansthatxisnottheadditiveidentityelement),thenF
containsanelementwhichwedenotex−1,whichisamultiplicativeinverseofx.Thismeans
thatmultiplyingxbyx−1givesthemultiplicativeidentityelement.
x×x−1 =1F (−x−1×x).
NOTE ON AXIOMS FM2 AND FM3: The multiplicative inverse acts like a reciprocal and it is
this that enables division (by non-zero elements) in a field. Dividing by an element x means
multiplying by x−1. The effect of multiplying by x−1 “cancels” the effect of multiplying by x.
Multiplying an element both by x and by x−1 has no effect - it amounts to multiplying by the
identityelement.
TERMINOLOGY
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1. Aringwhosemultiplicatoniscommutativeisreferredtoasacommutativering.
2. A ring that has an identity element for multiplication is referred to as a unital ring or as a
ringwithidentity.
3. Inaringwithidentity,wecanconsiderwhichelementshavemultiplicativeinverses. These
elementsarecalledtheunitsofthering.
4. A ring in which every non-zero element is a unit is called a division ring. A division ring
satisfies all the axioms of a field except for the commutativity of multiplication. We will
meetanexampleofanon-commutativedivisionringlater.
EXERCISES
1. WhichoftheexamplesfromLecture1arecommutative?
2. WhichoftheexamplesformLecture1areringswithidentity? Foreachsuchexample,state
whatthemultiplicativeidentityelementis.
3. InthoseexamplesfromLecture1thatarehaveanidentityelementformultiplication,state
whichelementsareunits. (Thisistrickierthanitmightappearatfirstglance).
4. WhataretheunitsintheringM (Z)of2×2matriceswithintegerentries?
2
The following lemma explains why Axiom FM3 only deals with non-zero elements. It says that
it is a consequence of the ring axioms that multiplying any element by the zero element will
always result in the zero element - so the zero element in a ring never has a chance of having a
multiplicativeinverse. Thislemmaisalsoaniceexampleofthesortofreasoningfromaxiomatic
foundationsthatiscentraltoabstractalgebra.
Lemma2.1. SupposethatrisanyelementofaringR. Then
0R×r=0Randr×0R =0R.
Proof. Wedealwiththeproduct0R×r. Becauseadding0Rtoanyelementhasnoeffect,weknow
that0R+0R =0R. Then
0R×r=(0R+0R)×r.
NowwecanusethedistributiveaxiomD2torewrite(0R+0R)×ras(0R×r)+(0R×r)andsay
0R×r=(0R×r)+(0R×r).
So we are saying that the element 0R ×r (which we would like to show is equal to 0R) remains
thesamewhenaddedtoitself. Nowwecanusesubtraction(orAxiomA5)tosimplifytheabove
equation.
(0R×r)−(0R×r) = (0R×r)+(0R×r)−(0R×r)
=⇒0R = 0R×r
asrequired. Theproofforr×0Rissimilar,butusesAxiomD1insteadofD2.
Note that proving this statement (that multiplying by zero always results in zero) uses quite a
fewoftheringaxioms. Inparticularitusesthedistributiveaxioms. Thesedescribehowaddition
and multiplication interact. What is special about the zero element is concerned with addition,
butwhatwearetryingtoproveissomethingabouthowthiselementbehavesundermultiplica-
tion. Soitisnotsurprisingtoseetheaxiomsdescribinghowadditionandmultiplicationinteract
enteringthepicture.
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3 Polynomials and Roots
Thissectionfocussesonpolynomialrings,especiallyonthepolynomialringQ[x]. Thisringhas
many structural properties in common with the ring Z of integers. For example both of these
ringsarecommutativeandeachcontainsanidentityelementformultiplication.
Wefinishedthelastsectionbyobservingthatinaring,multiplyingbythezeroelementalways
resultsinthezeroelement. Onecouldasktheconversequestion: iftheproductoftwoelements
inringiszero,mustone(orboth)ofthetwoelementsbezero?
Theanswertothisquestionisnotalwaysyes.
Exercise: Giveanexampleofapairofnon-zeroelementsofD (Z)whoseproductiszero.
2
However,formanyringstheanswerisyesandwemakefrequentuseofthisfact.Forexample,
whenwesolvethequadraticequationx2+5x+6=0bywriting
x2+5x+6=0=⇒(x+2)(x+3)=0=⇒x+2=0orx+3=0,
wearemakinguseofthefactthattheproductoftworealnumbersis0ifandonlyifatleastone
ofthenumbersis0.
DefinitionAnintegraldomainRisacommutativeringwithidentityinwhichtheproducta×bis
equaltothezeroelement0Ronlyifeithera=0Rorb=0R.
Example The ring Z of integers is an integral domain. It is from the term “integer” that the
terminology“integraldomain”arose. ThefieldQofrationalnumbersisalsoanintegraldomain.
Infactallfieldsareintegraldomains.
The ring Q[x] is another example of an integral domain. Certainly it is a commutative ring
with identity. What is required to see that it is an integral domain is a demonstration that the
productoftwonon-zeropolynomialsinQ[x]cannotbezero. Sosupposethatp(x)andq(x)are
non-zero polynomials in Q[x] and let their respective leading terms be axt and bxs. This means
thataandbarenon-zerorationalnumbers,andthatnohigherpowerofxthanxt appearswith
non-zerocoefficientinp(x),andnohigherpowerofxthanxs appearswithnon-zerocoefficient
inq(x). So
p(x)=axt+possiblelowerordertermsinx, q(x)=bxs+possiblelowerordertermsinx
Thentheleadingtermoftheproductp(x)q(x)isabxt+s: weknowthatab (cid:54)= 0sinceaandbare
non-zerorationalnumbers.Notethatweareusingthefactthatonlytheleadingtermsofp(x)and
q(x)cancontributetoatermofordert+sintheproduct, soxt+s isthehighestpowerofxthat
canappearinp(x)q(x),anditappearstherewithcoefficientab. Finally,sincewehaveidentified
atleastonenon-zeroterminp(x)q(x),wecanassertthatp(x)q(x) (cid:54)= 0andhencethatQ[x]isan
integraldomain.
Notes
1. Itisperfectlypossibleforeithertors(orboth)tobeequaltozerointheargumentabove
- this justmeans thateither p(x) or q(x) (orboth) isa non-zeroconstant polynomial. This
“extremecase”causesnoproblemsintheproofabove.
2. A piece of terminology: the highest power of x to appear with non-zero coefficient in a
polynomialp(x)iscalledthedegreeofp(x). Thedegreeofanon-zeroconstantpolynomial
is0. (Thedegreeofthezeropolynomialisnotdefined). Ifp(x)andq(x)aretwonon-zero
polynomials in Q[x], then the degree of the product p(x)q(x) is equal to the sum of the
degreesofp(x)andq(x).
Exercises
1. Provethateveryfieldisanintegraldomain.
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