Vol. 6_1_ Cont. Appl. Sci. 52-62

8/4/2019 Vol. 6_1_ Cont. Appl. Sci. 52-62

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52

Continental J. Applied Sciences 6 (1): 52 - 62, 2011 ISSN: 1597 - 9928© Wilolud Journals, 2011 http://www.wiloludjournal.com

Printed in Nigeria

TTHHEE HHIILLBBEERRTT’’SS NNUULLLLSSTTEELLLLEENNSSAATTZZ AANN DD IITTSS AAPPPPLLIICCAATTIIOONN TTOO TTHH EE SSTTUUDDYY OO FF AALLGG EEBBRRAAIICC CCUU RRVV EESS-- AA SSUURRVV EEYY

II..KK.. DDoonnttwwii,, SS..AA.. OOppook k uu aanndd WW ..OObbeenngg--DD eenntteehh DDeeppaarrttmmeenntt oof f MM aatthheemmaattiiccss,, KK wwaammee NN k k rruummaahh UUnniivveerrssiittyy oof f SScciieennccee aanndd TTeecchhnnoollooggyy,, KKuumm aassii,, GGhhaannaa

AA BBSSTTRRAACCTT AA llggeebbrraaiicc GGeeoommeettrryy iiss oonnee oof f tthhee oollddeesstt oof f tthhee ccllaassssiiccaall mmaatthheemmaattiiccaall ddiisscciipplliinneess.. TTrraaddiittiioonnaallllyy,, ii tt ddeeaallss ww iitthh tthhee ssttuuddyy oof f aallggeebbrraaiicc vvaarriieettiieess,, tthhaatt iiss,, tthhee zzeerroo--sseettss oof f ssyysstteemm ss oof f ppoollyynnoommiiaall eeqquuaattiioonnss,, iinn ppaarrttiiccuullaarr,, ww iitthh tthheeiirr ggeeoomm eettrryy.. TThhee iimmppoorrtt oof f tthhiiss ww oo rr kk ii ss ttoo f f iinndd tthhee ppaannaacceeaa ttoo tthhee ssttuuddyy oof f AAllggeebbrraaiicc ccuurrvveess bbyy

aappppllyyiinngg tthhee HH iillbbeerrtt’’ss NNuullllsstteelllleennssaattzz.. TThhiiss iiss ttoo aallllooww f f oorr f f uullll ggrraasspp oof f tthhee tthheeoorryy oof f aallggeebbrraaiicc ccuurrvveess.. TThhee NN uullllsstteellIIeennssaattzz rreellaatteess vvaarriieettiieess aanndd iiddeeaallss iinn ppoollyynnoommiiaall rriinnggss oovveerr aallggeebbrraaiiccaallllyy cclloosseedd f f iieellddss.. TThhee rreessuulltt iiss tthhaatt tthheerree iiss aa oonnee--ttoo--oonnee ccoorrrreessppoonnddeennccee bbeettwweeeenn rraaddiiccaall iiddeeaallss aanndd aallggeebbrraaiicc sseettss.. TThhiiss hhaass lleedd ttoo mmaannyy rreessuullttss aanndd aa ddeeeeppeerr uunnddeerrssttaannddiinngg oof f tthhee ccllaassssiiccaall tthheeoorreemm --HHiillbbeerrtt’’ss NNuullllsstteelllleennssaattzz..

KEYWORDS: Hilbert’s Nullstellensatz, algebraic curves, ideals, variety, modules

IINN TTRROODD UUCCTTIIOONN Taking into account Algebraic geometry, it goes without saying that although it is a highly developed, persisting andthriving edifice of Mathematics, it is famously intricate for the novice to make headway in the subject. Thisenterprise was therefore embarked on, that is the Application of the Hilbert’s Nullstellensatz to the study of Algebraic curves.

Fig. 1 Picture of David Hilbert

David Hilbert was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk,near Kaliningrad, Russia) on January 23,1862 and his demise occurred on February 14, 1943, who isrecognized as one of the most influential mathematicians of the 19th and early 20th centuries. His owndiscoveries alone would have given him that honour, yet it was his leadership in the field of mathematics

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DDoonnttwwii et al.,: Continental J. Applied Sciences 6 (1): 52 - 62, 2011

throughout his later life that distinguishes him. He held a professorship in mathematics at the University of Göttingen for most of his life and this was appreciable in its real sense of the word (Biography of DavidHilbert).The picture of David Hilbert can be seen at Fig.1.

He solved the principal problem in nineteenth century invariant theory by showing that any form of a givennumber of variables and of Hilbert's basis theorem in a given degree has a finite, yet complete system of independent rational integral invariants and covariants (Exampleproblems,17-02-2011). Hilbert solved severalimportant problems in the theory of invariants, algebraic number theory with his 1897 treatise Zahlbericht whichliterally meant "report on numbers". Hilbert also solved Gordon’s problem.

OObb j j eeccttiivvee oof f SSttuuddyy Many students struggle with algebra through no fault of their own. After all, algebra has its own uniquelanguage and set of rules. Nonetheless, the frustration experienced by algebra students can result in a loss of self-confidence.

In the quest to find the panacea to the issues raised, the following objective has been set. The aim of this work

is to develop the theory of algebraic curves from the viewpoint of modern algebraic geometry but withoutexcessive prerequisites and based on the Hilbert’s Nullstellensatz as far as possible (Obeng-Denteh, 2006).

AAllggeebbrraaiicc GGeeoommeettrryy iiss oonnee oof f tthhee oollddeesstt oof f tthhee ccllaassssiiccaall mmaatthheemmaattiiccaall ddiisscciipplliinneess. TTrraaddiittiioonnaallllyy,, ii tt ddeeaallss wwiitthh tthhee ssttuuddyy oof f aallggeebbrraaiicc vvaarriieettiieess,, tthhaatt iiss,, tthhee zzeerroo--sseettss oof f ssyysstteemmss oof f ppoollyynnoommiiaall eeqquuaattiioonnss,, iinn ppaarrttiiccuullaarr,, wwiitthh tthheeiirr ggeeoommeettrryy.. In his paper Harris (1980) sets the problem of finding what may be the Hilbert function of a generichyperplane section of a reduced irreducible curve.

It goes without telling that Polynomial equations have a lengthy history. In 1494 Pacioli ended his Summa diArithmetica with the remark that the solution of the equations x 3 + mx =n and x 3 + n = mx was as impossible atthe existing state of knowledge as squaring the circle (Rideout, D.,-----) All the formulae discovered had onestriking property, which can be illustrated by Fontana’s solution of x 3 + px = q:

3 23

2 27 4q p q

x = + + +3 2

3

2 27 4q p q

− +

Polynomial equations f(x) = 0 (where f is a polynomial) have been of considerable historical importance inmathematics. For any given polynomial f the problem of finding a solution, or even all of them, is one of computation; but a general theory requires more refined techniques.

When a curve is not algebraic, we call the curve and its function transcendental.In the case a function is sufficiently sophisticated it is said to be a special function (Special function, 02-04-11).

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in

general was quite fully developed in the nineteenth century, after many particular examples had beenconsidered, starting with circles and other conic sections.

Fig 2 :The Tschirnhausen cubic is an algebraic curve of degree three.

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An algebraic curve defined over a field F may be considered as the locus of points in F n determined by at leastn − 1 independent polynomial functions in n variables with coefficients in F, g i(x1, …, x n), where the curve isdefined by setting each g i = 0.

Using the resultant, we can eliminate all but two of the variables and reduce the curve to a birationallyequivalent plane curve, f(x, y) = 0, still with coefficients in F, but usually of higher degree, and often possessingadditional singularities. For example, eliminating z between the two equations x 2 + y 2 − z 2 = 0 andx + 2y + 3z − 1 = 0, which defines an intersection of a cone and a plane in three dimensions, we obtain the conicsection 8x 2 + 5y 2 − 4xy + 2x + 4y − 1 = 0, which in this case is an ellipse. If we eliminate z between4x 2 + y 2 − z 2 = 1 and z = x 2, we obtain y 2 = x 4 − 4x 2 + 1, which is the equation of a hyperelliptic curve(Algebraic curve, wiki).

In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraicvarieties are one of the central objects of study in algebraic geometry (Algebraic geometry, Wiki). The word"variety" is employed in the sense of a manifold, for which cognates of the word "variety" are used in theRomance languages.

Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra andgeometry by showing that a monic polynomial in one variable with complex coefficients i.e., an algebraicobject, is determined by the set of its roots, i.e., a geometric object. Building on this result, Hilbert'sNullstellensatz provides a fundamental correspondence between ideals of polynomial rings and subsets of affinespace. Using the Nullstellensatz and related results, mathematicians are able to capture the geometric notion of avariety in algebraic terms as well as bring geometry to bear on questions of ring theory (Algebraic Variety,wiki).

In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.For example, for a set {f i} of polynomials over a ring R, the solution set is the subset of R on which thepolynomials all vanish (evaluate to 0), formally (Solution set,Wikipedia)

RRiinnggss:: DDeef f iinniittiioonn:: AA rriinngg iiss aa sseett RR ttooggeetthheerr wwiitthh ttwwoo bbiinnaarryy ooppeerraattiioonnss nnaammeellyy aaddddiittiioonn ( )+ aanndd mmuullttiipplliiccaattiioonn ((xx)) ddeef f iinneedd oonn iitt ssuucchh tthhaatt tthheessee ccoonnddiittiioonnss aarree ssaattiissf f iieedd..

11.. FFoorr eevveerryy ppaaiirr Rba ∈, bbootthh oof f ba + aanndd ba * bbeelloonngg ttoo RR.. WW hheenn tthheerree iiss nnoo aammbbiigguuiittyy aabboouutt

tthhee mmuullttiipplliiccaattiioonn tthheenn baab *=

22.. AAddddiittiioonn iiss ccoommmmuuttaattiivvee ii..ee.. abba +=+ f f oorr eevveerryy ppaaiirr ., Rba ∈

33.. TThheerree eexxiissttss Ro ∈ ssuucchh tthhaatt aoa =+ f f oorr aallll Ra ∈

44.. FFoorr eevveerryy R x ∈ tthheerree eexxiissttss R x∈− ssuucchh tthhaatt .o x x =+−

55..

( ) ( )cbacba++=++

wwhheenneevveerr ,,, Rcba∈

aaddddiittiioonn iiss aassssoocciiaattiivvee.. IInn vviieeww oof f tthhiiss iitt iiss aann aacccceepptteedd nnoorrmm tthhaatt ( ) ( )a b c a b c a b c+ + = + + = + +

AAxxiioommss 5,4,3,2 mmaak k ee R aann AAbbeelliiaann ggrroouupp uunnddeerr aaddddiittiioonn ((OObbeenngg--DDeenntteehh,, 22000066;; OOppook k uu,, 22000044))

( ) ( ) cbacba **** = wwhheenneevveerr Rcba ∈,, [[ii ..ee.. ** iiss aassssoocciiaattiivvee]]

IInn vviieeww oof f tthhiiss iitt iiss aann aacccceepptteedd nnoorrmm tthhaatt ( ) ( ) cbacbaabc **** == wwhheenneevveerr .,, Rcba ∈

66.. ( ) ( ) ya xa y xai *** +=+ wwhheenneevveerr R y xa ∈,, [[ii ..ee.. ** iiss ddiissttrriibbuuttiivvee oovveerr ++]]..

( )( ) a ya xa y xii *** +=+ wwhheenneevveerr .,, R y xa ∈

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( ) ( )( ) ( ) ( )( )k cbcbd ad a

jd bcacabd id cd cbabad d ccbbbak d jciba xk d jciba y x

kj jk kiik

12211221

212112211221122121212111

2222,,,,.

,

−+++

−+++−+++−−−=

++++++=

≠≠

TThhiiss pprroodduucctt f f oorrmmuullaa iiss oobbttaaiinneedd bbyy mmuullttiippllyyiinngg tthhee f f oorrmmaall ssuummss tteerrmm bbyy tteerrmm ssuubb j jeecctt ttoo tthhee f f oolllloowwiinngg rreellaattiioonnss ((OObbeenngg--DDeenntteehh,, 22000066;; OOppook k uu,, 22000044))::

AAssssoocciiaattiivviittyy;; ir ri = ;;

kr rk jr rj == ; ((f f oorr aallll ) Rr ∈ ;; ;1222 −==== ijk k jl

jkikiikj jk k jiij ===−=−=−= ;

AAnnootthheerr iimmppoorrttaanntt eexxaammppllee oof f aa rriinngg ::LLeett K bbee aa rriinngg wwiitthh uunniittyy 11..AA ppoollyynnoommiiaall f f oovveerr KK iinn iinnddeetteerrmmiinnaattee X

iiss aa f f oorrmmaall ssuumm f f == ∑∞

= 0n

nn xa wwhheerree K a n

< f f oorr aallll 0≥n aanndd tthheerree eexxiissttss aa ppoossiittiivvee iinntteeggeerr K ssuucchh tthhaatt

0=na f f oorr aallll qn ≥ .. [ ] X K ddeennootteess tthhee sseett oof f aallll ppoollyynnoommiiaallss oovveerr K iinn iinnddeetteerrmmiinnaattee X .. FFoorr

0∑

∞

=

=

n

nn xa f aanndd

0

d∑∞

=

=

n

nn xh iinn [ ] X K .. UUssuuaallllyy h f = iif f aanndd oonnllyy iif f nn d a = f f oorr

0≥n .. [ ] X K aallssoo hhaass tthhee ssttrruuccttuurree oof f aa

ccoommmmuuttaattiivvee rriinngg wwiitthh uunniittyy 11 aass f f oolllloowwss::

FFoorr eevveerryy ppaaiirr 0

∑∞

=

=

n

nn xa f

AAnndd ∑∞

=

=

0n

nn xbg iinn [ ] X K

( )∑∞

=

+=+

0nnn and bag f

( )

nn

n

nqqn

n

qn

xc fg

xbag f

∑

∑∑∞

=

−

=

∞

=

=

=

0

00

.

WW hheerree 00

≥=−

∞

=

∑ nevery for bac qqnq

n

nq j

nq jn

xba fg

= ∑∑

=+

∞

= 0

LLeett ∑∞

==

0n

nn xa f bbee aa

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PPoollyynnoommiiaall oovveerr KK iinn iinnddeetteerrmmiinnaattee XX..

AA.. IIf f 0=na f f oorr aallll

gg f thenn=+≥

0 f f oorr aallll ].[ X K g ∈

TThheerreef f oorree iinn tthhiiss ccaassee f f iiss tthhee zzeerroo iinn ][ X K aanndd f f iiss ccaalllleedd aa ppoollyynnoommiiaall oof f ddeeggrreeee -- ∞ oovveerr K ..

BB.. IIf f 000=≠

naand a f f oorr aallll 0 1 a f thenn =≥ aanndd iinn tthhiiss ccaassee wwhheenn g f = aanndd 00≠a

tthheenn f f iiss aa ppoollyynnoommiiaall oof f ddeeggrreeee 00..

CC.. IIf f q iiss aa ppoossiittiivvee iinntteeggeerr 00 =≠nq aand a f f oorr aallll 1+≥ qn

TThheenn q

q xa xaa f +++= .....10

IInn tthhiiss ccaassee f f iiss ccaalllleedd aa ppoollyynnoommiiaall oof f ddeeggrreeee qq.. 32

212 3 , x x xg x f +++==

543221

3221

3221

32

3

4

3

000

x x x x fg

x x xg f

x x xg

x x x f

+++=

+++=+

+++=+

+++=

DDeef f iinniittiioonn:: AAnn eelleemmeenntt ][ X K f ∈ oof f ddeeggrreeee 0≥q iiss ssaaiidd ttoo bbee mmoonniicc:: iif f 1=

qa

IIddeeaallss:: LLeett R bbee aa rriinngg.. AA ssuubbsseett I oof f R iiss ccaalllleedd aann iiddeeaall iinn R iif f tthheessee ccoonnddiittiioonnss aarree ssaattiissf f iieedd::

11.. I ∈0

22.. FFoorr eevveerryy Rr ∈ aanndd eevveerryy I a ∈ bbootthh oof f ar ra and bbeelloonngg ttoo I

33.. FFoorr eevveerryy ppaaiirr I ba ∈, ,, aa –– bb ∈∈ II

EExxaammppllee:: LLeett K bbee aa ccoommmmuuttaattiivvee rriinngg

11.. GGiivveenn K a ∈ lleett { }Ka / ∈= λ λ aka

TThheenn Ka iiss aann iiddeeaall iinn K aanndd Ka iiss tthhee pprriinncciippaall iiddeeaall iinn K ggeenneerraatteedd bbyy a ..

22.. GGiivveenn aa nnoonn--eemmppttyy ssuubb j j eecctt X oof f K .. LLeett )( X S bbee tthhee sseett oof f aallll f f iinniittee ssuummss ∑=

m

j j j x

1

λ wwhheerree

K m∈}.,.........{ 1 λ λ aanndd X x x m

∈},.........{ 1

TThheenn )( X S iiss aann iiddeeaall iinn K .. IItt iiss ccaalllleedd tthhee iiddeeaall iinn K ggeenneerraatteedd bbyy X ..

PPrroooof f :: CChhoooossee X u ∈ .. TThheenn )(00 X Su ∈=

IIf f K r ∈ aanndd )( X Sa ∈ cchhoooossee f f iinniitteellyy mmaannyy eelleemmeennttss n y y ,.......1 ssuucchh

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tthhaatt =a ∑=

∈

n

jn j j K where y

11 ,........γ γ γ

TThheenn ( ) j j

n

j yr ar ra γ ∑=

==1

( )∑=

∈=n

j j j X S yr

1

)(γ

FFiinnaallllyy,, iif f ,.....1 nγ γ

K ww y y

and K

qn

q

∈

∈

..................

.........

11

1

1

η η

SSuucchh tthhaatt t

n

jt ya ∑

=

=

1

γ aanndd ∑=

=q

t t t wb

1

η ..TThheenn ∑ ∑= −

∈−=−

1 1

)( j t

t t j j X Sw yba η γ

iiss aann iiddeeaall iinn K ..

MM oodduulleess:: LLeett R bbee aa rriinngg AA.. MM oodduullee M oovveerr R iiss aa sseett M ttooggeetthheerr wwiitthh ttwwoo pprroocceesssseess ddeef f iinneedd aass f f oolllloowwss::

FFoorr eevveerryy ppaaiirr M y x ∈, M y x ∈+

FFoorr eevveerryy R∈λ aanndd eevveerryy M x∈ aanndd tthheessee ccoonnddiittiioonnss aarree ssaattiissf f iieedd::

11.. x y y xi +=+)( f f oorr eevveerryy M y x ∈,

z y x z y xii ++=++ )()()( wwhheenneevveerr M z y x ∈,,

)(iii TThheerree eexxiissttss M ∈0 ssuucchh tthhaatt x x =+0 f f oorr eevveerryy M x∈ )(iv FFoorr eevveerryy M x∈ tthheerree eexxiissttss 0=+− x x

22.. FFoorr eevveerryy ppaaiirr M y x ∈, aanndd eevveerryy R∈λ

y x y x λ λ λ +=+ )( ..

33.. WW hheenn R iiss aa rriinngg wwiitthh uunniittyy 1 aanndd x x =1 ,,f f oorr eevveerryy M x∈ tthheenn M iiss ccaalllleedd aa uunniittaarryy mmoodduullee..

44.. FFoorr eevveerryy ppaaiirr R∈ µ λ , aanndd eevveerryy M x∈

.))((

)())((

x x xii

x xi

µ λ µ λ µ λ λµ

+=+

=

DDeef f iinniittiioonn:: LLeett F aanndd K bbee f f iieellddss.. K iiss ssaaiidd ttoo bbee aa f f iieelldd eexxtteennssiioonn oof f F ((oorr ssiimmppllyy K iiss aann eexxtteennssiioonn oof f F )) iif f F iiss aa ssuubbf f iieelldd oof f K ..

EExxaammpplleess:: LLeett QQ bbee f f iieelldd oof f aallll rraattiioonnaall nnuummbbeerrss,, RR tthhee f f iieelldd oof f aallll rreeaall nnuummbbeerrss aanndd CC tthhee f f iieelldd oof f aallll ccoommpplleexx nnuummbbeerrss ((OObbeenngg--DDeenntteehh,, 22000066;; OOppook k uu,, 22000044;; OOppook k uu aanndd DDoonnttwwii,, 22000000))..

TThheenn 11.. RR iiss aann eexxtteennssiioonn oof f QQ.. 22.. CC iiss aann eexxtteennssiioonn oof f RR .. 33.. CC iiss aann eexxtteennssiioonn oof f QQ ..

44..

EExxaammpplleess:: 55.. LLeett k k == RR..

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Affine VarietiesWe define A n(k) or more simply A n, when k is clear from the context, called the affine n-space over k, to be k n.The purpose of this notation is to emphasize that one 'forgets' the vector space structure that k n carries.

Abstractly speaking, An

is, for the moment, just a collection of points.

A function f : A n → A1 is said to be regular if it can be written as a polynomial, that is, if there is a polynomial pin k[x 1,...,x n] such that f(t 1,...,t n) = p(t 1,...,t n) for every point (t 1,...,t n) of A n.

Let R be a field and consider the systems of equationsx + y + z = 1 and x + 2y - z = 3

Geometrically, this represents the line R 3 which is the intersection of the planes x + y + z = 1 and x + 2y - z= 3.

It follows that there are infinitely many solutions (Cox et al, 2007).

Regular functions on affine n-space are thus exactly the same as polynomials over k in n variables. We will referto the set of all regular functions on A n as k[A n].

We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k[A n]. The vanishing set of S (or vanishing locus) is the set V(S) of all points in A n where everypolynomial in S vanishes. In other words,

A subset of A n which is V(S), for some S, is called an algebraic set. The V stands for variety.

In the two scenarios below:Given a subset U of A n, when is U = V(I(U))?Given a set S of polynomials, when is S = I(V(S))?

The answer to the first question is provided by introducing the Zariski topology ( ZZaarriissk k ii ,, 11998822;; Algebraicgeometry, 02-04-11), a topology on A n which directly reflects the algebraic structure of k[A n]. Then U =V(I(U)) if and only if U is a Zariski-closed set. The answer to the second question is given by Hilbert'sNullstellensatz. In one of its forms, it says that I(V(S)) is the prime radical of the ideal generated by S. In moreabstract language, there is a Galois connection, giving rise to two closure operators; they can be identified, andnaturally play a basic role in the theory; the example is elaborated at Galois connection (Algebraic geometry,02-04-11).

Fig. 3 Examples of Algebraic Curves

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Obeng-Denteh,W. (2006): The Hilbert’s Nullstellensatz And its Application to the Study of Algebraic Curves,

MSc thesis , DDeeppaarrttmmeenntt oof f MM aatthheemmaattiiccss,, KK wwaammee NNk k rruummaahh UUnniivveerrssiittyy oof f SScciieennccee aanndd TTeecchhnnoollooggyy,, KK uummaassii,, GGhhaannaa,,pppp 2277 –– 3366..

OOppook k uu,, SS..AA..,, aanndd DDoonnttwwii,, II ..KK.. ((22000000)):: TTooppoollooggyy,, LLeeccttuurree NNootteess f f oorr UUnnddeerrggrraadduuaattee SSttuuddeennttss,, MM aatthheemmaattiiccss DDeeppaarrttmmeenntt,, KKwwaammee NNk k rruummaahh UUnniivveerrssiittyy oof f SScciieennccee aanndd TTeecchhnnoollooggyy,, KKuummaassii,, GGhhaannaa..

OOppook k uu,, SS..AA.. ((22000044)):: LLeeccttuurree nnootteess oonn AAbbssttrraacctt AAllggeebbrraa f f oorr PPoossttggrraadduuaattee ssttuuddeennttss,, MM aatthheemmaattiiccss DDeeppaarrttmmeenntttt,, KKwwaammee NNk k rruummaahh UUnniivveerrssiittyy oof f SScciieennccee aanndd TTeecchhnnoollooggyy,, KKuummaassii,, GGhhaannaa,,11--1155,,3300--3355

OOppook k uu,, SS..AA.. ((22000055)):: LLeeccttuurree nnootteess oonn GGaallooiiss TThheeoorryy f f oorr PPoossttggrraadduuaattee ssttuuddeennttss,, MM aatthheemmaattiiccss DDeeppaarrttmmeenntt,, KKwwaammee NNk k rruummaahh UUnniivveerrssiittyy oof f SScciieennccee aanndd TTeecchhnnoollooggyy,, KKuummaassii,, GGhhaannaa,, pppp..11--3333,,4455--6611

RRiiddeeoouutt,, DD.. ((----------)):: SSoolluuttiioonnss oof f PPoollyynnoommiiaall EEqquuaattiioonnss,, MM eemmoorriiaall UUnniivveerrssiittyy oof f NNeewwf f oouunnddllaanndd

Solution set, Retrieved on 02-04-2011from http://en.wikipedia.org/wiki/Solution_set

Special function, Retrieved on 02-04-2011 fromhttp://www.2dcurves.com/algebraic.html http://en.wikipedia.org/wiki/Algebraic_curve

ZZaarriissk k ii ,, OO.. ((11998822)):: AAllggeebbrraaiicc SSyysstteemmss oof f PPllaannee CCuurrvveess,, AAmmeerr JJ.. MM aatthhss 110044..

Received for Publication: 06/04/11Accepted for Publication: 07/05/11

Corresponding AuthorWW ..OObbeenngg--DDeenntteehh

DDeeppaarrttmmeenntt oof f MM aatthheemmaattiiccss,, KKwwaammee NNk k rruummaahh UUnniivveerrssiittyy oof f SScciieennccee aanndd TTeecchhnnoollooggyy,, KKuummaassii,, GGhhaannaa EEmmaaiill:: oobbeennggddeenntteehhww@@ yyaahhoooo..ccoomm

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Vol. 6_1_ Cont. Appl. Sci. 52-62