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The algebraic number realms K ([square root] -2), K ([square root] 5) and K ([square root] -23) Fisher, Mary Jean 1931

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THE ALGEBRAIC NUMBER REALM Sfa • by Mary Jean Fisher A Thesis submitted for the Degree of MASTER OF ARTS ±11 the Department of MATHEMATICS The University of British Columbia January - 1931. Table of Contents, page Introduction, I Part I The Realm M f ^ a ) . 1 Part II The Realm -Hinr) . 14 Part III The Realm -Hit S3). 2 6 w Mtt)* -k(fT) M D MiT=*3) Introduction in algebraic number is a number which satisfies a ration-al equation, via,, an equation of the form where Cb, >CLjl) ^¿^are rational numbers, In algebraic integer is an algebraic number which satis-fies an equation of the form 1} in which CLMj A«. are rational integers. Hereafter, algebraic numbers and algebraic integers will usually be referred to as numbers and integers. If equation 1} is the rational equation of lowest degree which a number cC satisfies, then 1) is irreducible, and oC €k is said to be a number of the n - degree, S^uation 1} is then called the rank or minimum equation of oC , in algebraic number realm, or, more briefly, a realm, is a system of algebraic numbers such that the sum, difference, product and quotient of any two numbers of the system, exclu-ding division by 0, are numbers of the system. Hence, if ©c is an algebraic number, the system consisting of all rational functions of cC , with rational coefficients, is a realm. Such a realm is denoted by , and «C is said to define 1) xn-+a.,xn~+ ==• 0. gfihf If oC is a number of the second degree, then -460 li said to be a realm of the second degree. In this case, the < / yeaaining root, << » of the equation defining «C , is called the conjugate of cC » a-ad is called the conjugate realm of -/iU) * I f th® realm 4(U) is identical with , it is called a Galois Realm. The purpose of this paper is to develop and contrast the algebraic number theory for the particular Galois realms and. . Vie find, for example, that the realms -fciO-¿J and -a3) have each only two units, viz., while the realm -ftLff) has an infinite number of units; also, that the Unique Factorization Theorem holds for the realms and t 5)» but does not hold for the realm Unique Factorization may, however, be restored in the latter realm by the introduction of ideals. V/e show that in the realm ~a)t the norm of a prime is either g , or a rational prime of the form / , or 8-n-*-3 . Likewise, in the realm -ft(J f)the norm of a prime is found to be or a rational prime of the form fit 2 / . In each of these realms the norm of the general integer appears as a binary quadratic form. With reference to these forms we derive theorems on representation, ¿m application of these theorems is made in obtaining solutions of certain types of Diophantine Equations. Corresponding results are not obtained for the realm -^YV * in which factorization is not unique. 1. PART I THE REALM 1» The lumbers of the Realm -¿¿if^ f). The number is defined by the equation 1} X*-h£ = O-Since ty —¿1 9 every number of ^¿jPS)» "the form where are rational numbers. Rationalising the de-nominator we obtain dC = A'^+zAf. + aA-a,,^, jZ* Hence every number of &(\J-jt) is of the form c( sa ¿t + where ¿2. arid -¿'are rational numbers. The other root of 1) defines the realm AkPsi) con jugate to -a) , But these realms ¿¡.re identical since -flLJ-S) contains all the numbers of -fcC-i)^ • contains ail the numbers of -k(\l-9i) • the number obt^ineu. The Conjugate end L'orm of a number of -fe^i/ -a), If^CsrtfWi^is a number o:C -^¿tPa), from <C by replacing yp^hj iod conjugate-/^ is called tho conjugate of cC • If «C = -f-4,and i^are t o number of -fcl\r~&) then ^ and hence* the conjugate of a product of two or more numbers of is equal to the product of the conjugates of its factors. The norm of a number of a realm of the second degree is the product of the number by its conjugate. The norm of << is, therefore, nM= C a W ^ - ^ t a h * - * 1 ! It follows that the norm of every number of -fclifcH) is a ¿posi-tive rational number. If «C an<^ a r e two numbers of -A(\f-jH) andeC'andj^ their conjugates, then ¿ci'-pp'^nW- ntfl. that is, the norm of a product of two or more numbers ox -¿l J is equal to the product of the norms of its factors. 3. Primitive and Imprimitive numbers of -2.). If A +-4is a number of -AiyT-z) then satisfies the equation a) y^-zax oy whose other root is oC ¿2-—4r » Thus every muuber of Pa) satisfies an equation of the second degree, The num-ber aC is said to be a primitive number of ^ if equ-ation 2) is irreducible, and to be iupri^itive if t-i.t e^udioxj. is reducible. Equation Z) is reducible if, and and socC is a primitive numbgr if, and. only ii; i. is uixxerent from its conjugate. The imprimitive numbers of -fit\f-a) therefore tlie rational numbers. 3. are4. Integers of -¿¿.J^g). A rational number is an algebraic integer when, and only when, it is a rational integer. Hence, of the imprimitive numbers of A U o n l y the rational integers are integers of -kitt)* The necessary and sufficient condition that a primitive number of an algebraic integer of H(\Pii) is that the coefficients of the rank equation of the number shall be ra-tional integers. Let a^-^/SbQ aii integer of -AitiZfi) • Then cC /-il is also aii integer of -A(ii-s) . The rank e^uaiions of << and «fi/s? are 3) * z-aa.,x 4, *= o, and 4) a Oj respectively. Therefore, from Z>) c*nd 4), Sd, *77t// a ^a«ionai integer; and Hence, we have 5) Substituting 5} in ."), 4 si >vf u rational L^t^er. i J 1 ! „ I t t V, Therefore 2ZL-#- is a rational integer, and we have %-*."' V ' ' JLf «2 » fj /. yi -&.0 yn^lil, a nd n ~ ö ö. Let n = then -rrf-h S'kf1^ O ynodtj, and w s - ^ s OynA^iO.' But Ä = and Therefore, if i* au integer / £ ' t At Af of A(J~Hr)> form -if'iC^ , where ¿i — u „ ra-tional integers. 5. liasis ox » ¿*ny two integers cuid a?^  of a. realm are ¿.aid lo form a "basis of the realm if every in^tgei of uhe realm can be ex-pressed in the form ¿¿^  ^ wlierö ^ ^ are r^liuuar integers. We have seen that every integer of -kbtt.) has the form O.-i-^ J^ si where ¿2. and -¿'are rational integers. Hence / and are a basis of •ACif^ ii) • The numbers H-^ -S. and ¿L+G/^ giiuay also be taken as a basis of -h(iPh) • To prove this, let be any integer of -hCitti) and suppose oC ss X (i-f Then X -f-ÄJ — ¿L x since and f^ are rational integers. — a rational integer 3a.—S a rational integer. Hence /-f tf-a and are a basis of Theorem. If and are a basis of JtiJ^a) the necessary and sufficient condition that where a r e integers, shall be also a basis of is a, a^ l - t l -The proof of the theorem for M f * > is exactly the same as that given for the realm -fc(c) in Held, n The Elements of the Theory of Algebraic lumbers", and so will not be repeated here. 6. The Discriminant of -frCtFUl). The discriminant of a realm is the squared determinant formed from any pair of basis numbers and their conjugates. The discriminant of therefore / / -f<i 7. Divisibility of Integers of -¿¿yP^l). An integer cC of MiPoJls said to be divisible by an in-teger^ of M f * ) if there exists an integer X of -¿¿¡Pa) such that kj'^t^i is divisible by f-il-2 since J+JiT:a - (l-f^X-i+F*). 8* The Units of -fifiT^ si)* Associated Integers. A unit of a realm is defined as an integer of the realm which divides every integer of the realm* Let £ = a unit of -kCJ^ii). Then £ must di-vide / and conversely every divisor of / is a unit. Hence c / where «c is an integer of )• . • X=* ± (f The units of ) are therefore jt Two integers, differing only in a unit factor, are said to be associated. The associates of any integer ©( therefore -/•«< and — << . 9. Prime lumbers of -fiiS^ iL)* An integer of AttPl) which is not a unit of iT^ a) and which has no divisors other than its associates and the units, is called a prime number of -A(O-sl)» To determine whether or not5 is a prime number of -ACiP*) we proceed as follows: Let $ iFaXc ), where e,d. are rational integers. Then either r ^ ^ l , 51 Lc* ^ tions ¿1 x -érs e -cLs. and is not a prime of Similarly it may "be shown that J and 3-f-ifsi etc., are primes of -kLf*)* 10. The Unique Factorization Theorem for -hLJ^ti)* The proofs of the three theorems, A, B, G, (below) upon which the proof of the Unique Factorization Theorem for -klìf-À) depends, and the proof of the Unique Factorization Theorem itself, for are identical with the proofs given for the corresponding theorems for the realm JlU) in Reid, "The Elements of the Theory of Algebraic numbers". Hence only the statement of the four theorems and their corollaries will be given. Theorem A. If cQ is an integer of -ftLiT*)and (3 is any in-teger of -kif^)different from 0 , there exists an integer,/*-of -fiiiFlt) such that Theorem B. If cC and {3 are any two integers of -ÀliT^ i) prime to each other, there exist two integers y and y of J-5) such that 8. , Goaf, 1» If ^  and f} are any two integers of -klitt) there ex-ists a common divisor ^ of ai and p such that every coram on divisor ofeC and p divides , and there exist two integers g and ^  of ~k(f-a) such that Gor. 2, If <tt)4*f ,*tnare any >v integers of if1*) there exists a common divisor J" of -vJ.such that every common divisor divides , and there exist yu in-tegers - jJn such that Theorem C. If the product of two integers < and (3 of ^CtT^i) is divisible "by a prime number V , at least one of the inte-gers is divisible by W . Gor, 1, If the product of any number of integers of -k(itt) is divisible by a prime number (T > at least one of the inte-gers is divisible by Tf. Gor, E, If neither of two integers is divisible "by a prime number fT , their product is not divisible by (T* Gor, 3, If the product of two integers ¿C and ft is divisible by an integer If , and neither nor ^  is divisible by X , then ^ is a composite number. The Unique Factorization Theorem. Every integer of can be represented in one, and only one, way as the product of prime numbers. Gor. 1, If oC and j3 are prime to each other, and Y is divisible by bothcC and , then X is divisible by their t V- Cor* S. If«< an* fl are each prime to y , then is prime to ar . Cor. 3. If«(, is prime to Y , and if <£(3 is divisible by y , ^ is divisible by . 11. Rational Prime Factors of Forms of Integers of ¿¿/^a). Theorem. A rational integer which may be represented by the norm of an integer of the realm i.e., by the form ', has its rational prime factors either (a) primes of the realm ¿CiF*). or (b) norms of primes of the realm. In case (a), these primes enter to even powers, and in case (b), the primes are themselves represented by the form * . Let Tf be any integer of -A/ife) , and write ff= f^ fFy • * where ff,,^, are primes of the realm, not necessarily distinct. Then Vi£Vr/ is a rational integer, and one of the type specified in the Theorem. Let n[rr]~ fAff'. . . • where fajpt> — - - are distinct rational primes, and ?, tZf positive rational integers. Then ft, divides Vi]jrfazn1 hence divides one of its rational prime factors, say . It cannot divide two such factors, for then it would divide their rational greatest common divisor, and hence would be a unit. Let 8} fr^SK* where is an integer of n[f>J= 7i[irJ~ njAj. Therefore, since we cannot have 7i[fr]=. / , we have either ryiCirJxjf From 9), , and it follows that fi/=.6L • Hence is the norm of a prime of -^//^¿J. From 10), it follows that is a unit, and hence that is a prime of A ( \ J • Since y xfrjz nfaJ nEfrJ' • • jot occurs to an even power in 7 .• IS. Representation "by the Binary Quadratic Form Lemma: The norm of a prime of -JkCiF-Si), not associated with a rational prime, is either 2 , or a rational prime of the form 9n+! or 8n+3 • The norm of a prime of ^t(f^i) which is as-sociated with a rational prime is the square of the rational prime, and is of the form 8n+ / • Rational primes of the forms 8h—I and 8/1 are primes of -¿¿if1.2) • Every rational prime of the form 8n+l or $n+$ is factorable into two con-jugate primes of -»f^ajand so is the norm of a prime of We notice that 9 and hence Si can be factored in-to two conjugate primes of -Aill-Si) , and so is not a prime of is til© norm of a prime of -A(tt) . /r=s be a prime of -¿¿¡Pa) % which is not associated with a rational prime, and consider fT ¿"/-a yiM a wliioh is» by inspection, congruent to -*•/ , or -t-3 , Let fr be a prime of -¿¿tPZ) , which is associated with a rational prime, say jp • Then /T= ¿y, , and ntnls.and we hare yiQrJs/» t W ^ • Rational primes of the forms 9n~t and are primes of , for a rational prime is factorable into conjugate primes of -Ahf^a) only if it is 3 , or of the form or Pn-f-3 . It remains to show that all primes of the forms and 971*3 are factorable into two conjugate primes of -A/iP*). In proof, if^S / or3 >7*4,9 , then the congruence ryuUt^ has solutions, since is a quadratic residue of all primes of the forms 8r\ + 1 and 9n + 3 * Let a, be a root. Then £tZ~ >n*dfi, and ((X -+ . But ¿L-t-1/-Si and ¿2 —/-a are integers of -fcCO Therefore, if yb is a prime of , yb must divide either ¿L-t-if-k or CL-lP^i • If a. ± \F~sl = fritttt) where C+diEzis an integer of ^kliPk) , then = ± / MUljwpjH'^  j ' - *t vV V ,.' * is impossible, sinoe fi and/¿are rational integers and jp > / • Hence is not a prime of "khT^) % "but is factorable into two conjugate primes of -A(if-si), and so is the norm of a prime of -AliTlt). As an immediate consequence of the lemma and the Theorem of Art* 11, we may state the following theorem on representa-tion by the binary quadratic form ¿¿Va^*. z /a Theorem. The binary quadratic form CL 't&'v , represents «? and all positive rational primes of the forms ^ny-/ and Pyl*+3 , and all positive rational integers which are pro-ducts of primes of these forms, and even powers of primes of the forms —/ and P n - 3 % In the latter case, the primes divide both a. and . The form of'-wir*cannot repre-sent positive rational primes of the form Sri"I or i or any positive rational integer which contains odd powers of primes of these forms. 13. The Diophantine Squat ions: Lx^-i-a^^t To find rational integral values of * and (j which satis-fy these equations, we have to find an integer <C of *AC\J-a) whose norm is the right member of the equations. If dCs/t+^ lfi-k then X st * tf— a r e solutions. Consider, first, the equation i) I. The integer £ satisfies this equation if, and only if, is a unit of -AU^i)* Hence ± J , and we have /, ^ & as the only solutions of i). t / * 13, next, the equation 11) wttere ^ i s a positive rational prime. By the Theorem of Art. 11, equation ii) has solutions if, and only if, or ynfid.8 • If 5 is an integer of -AtiT^) whose norm is Jo , then Xsr+4, , are the solutions of ii). Consider, finally, the equation iii) x ^ s L ^ y t L , where >7tis a positive rational integer. By the Theorem of Art. 11, equation iii) has solutions if, and only if, where ^«i»8^ ® distinct primes of the forms + l and tn+'Z ¿L ; • > d i s t i n c t primes of the forms and , and c,,l\tt — - - positive rational integers. Then if we have 11) . . « W - ' * Hence, , , are solutions of iii). By inter-changing one or more of the ^ with their conjugates in 11), we obtain all possible factorisation of minto conjugate fac-tors. Let 'Jbe the result of any such inter-•J «/ V change. Then » a r e solutions of iii). Pro-ceeding in this way, all the solutions of iii) may be obtained. f: 14. PART II THE REALM M i l ) !• The lumbers of the Realm 'hi\T6)* The number /¿T is defined by the equation 1) -Since =r CT , every number of ^¿V^is of the form t where are rational numbers. Ra-tionalizing the denominator, we see that every number of -kCiff) is of the form ¿t -h-tiTf, where ¿Z and -¿'are rational numbers. The other root of 1) defines the realm fif)* conjugate to . As in -ftCiFii) , the realms and AfSfJ are identical. &• The Conjugate and Xorm of a dumber of . If (3= is a number of , then is its conjugate. The conjugate of a product of two or more numbers of is equal to the product of the conjugates of its factors. The norm of fl— CL~f-*4r i/W is ytifil The norm of a product of two or more numbers of 15. Is equal to the product of the norms of its factors. Primitive and Imprimitive lumbers of -A/iisJ» Every number firs CL-t-JriTf of -ftlifF) satisfies the equation Z) X As in ACiTh) , if equation 2) is irreducible, the number (3 is said to be a primitive number of -Miff) , and if Z) is re-ducible, ft is said to be an imprimitive number of The imprimitive numbers of -klff) are the rational numbers. 4. Integers of . Of the imprimitive numbers of -A(fs) only the rational in-tegers are integers of Let ft-^+J; iff be an integer of -A/VTj . Then /^/F5 is also an integer of -foiiT^ ) • The rank equations of ft , and are 3) yc - 2afx -HI*- o, and 4) = O, respectively. Prom 3) and 4), — Vrvf a rational integer /¿^ = tt That is, u ir 5) Substituting in 3) , ft: A rv Id X ~~>7t.X -jfe =5 ¿> therefore is a rational integer, and we have fvnz-nxs.6 yrmJl-y ' . n ^ o y n t d f and y Let Then vyi*- /j ; i.e., since f&jyruML/j • Therefore yjuwia . Hence >n and -/f are both odd or both even. But VI— 4 , and = jjt> Hence every integer of is of the form where <t and are both odd or both even. Basis of The integers of AhiT) are of the form where ¿2, and •if'are both odd or both even. Hence / and fir are not a basis of -klfs) , but we may prove that / and IdEiOiL- , which are integers of -ALiTF), form a basis of -kliT?) . To do this, let oCzz be an integer of -hlSf) and suppose cC-X+J-L+fZ . Then X-i-!f/A-and = • Therefore » a rational integer; and a ration-al integer since CL and are both odd or both even. Hence, every integer of -Alif?) can be expressed in the form j-tjS-* where X and^ are rational integers. Therefore / and / are a basis of At\TF) • A theorem similar to that stated in Part rA/tT5) holds for ALifT). 17. I, Art. 5, for 6. The Discriminant of -ALifs). The discriminant of -kiiTf) is f/ L ^ * / 7. The Conjugate and ETorm of the Integers of -kCiTs) * Let / and a) , where (O ss. y^jTW f he a basis of «z Then co'~ » ancl eO+cfi'-z / , ¿^¿¿/s—/. Hence satisfies the equation * ' X — / »¿J -Let ^ss a-f-^o) be an integer of Then and (p.+~4'u>J(&-rM/-a>)} SL ¿L-t-CL -+~4rS<0 — i a rational integer since ^  and ^are rational integers. 8. Divisibility of Integers of-^¿VFA An integer «C of -Ali/l^ lls said to be divisible by an in-teger (3 of -ACiTJ) if there exists an integer ^ of -k(i/p") such that Thus 2. -f-2. / jr is divisible by 3 — /¿^ since RV 2 -i-s =-¿3- iTr) -9. The Units of -hLiTF)* Associated Integers. The norm of every integer of -A/SJ) which is a unit is jt/. Hence, every integer of the second degree which is a unit, satisfies an equation of the form 6) x*-t-a<£ To find the integers different from / which are units of the realm, let CL take on the values i I, If CL a: 3t / , equation 6} becomes either and has the solutions x s. " ! l v - +/±iT=T which are not integers of -hlt/s); or else and has the solutions V = ±jtlr y - ft * ^ . which ^ «2. are integers arid units of Jiti/IF)* When » 3etc., other units of "A(ifF)may be determined in a similar way. .Evidently /ifir is the smallest unit greater than / • Theorem. .all units of -faltis) have the form ± , where H is a positive or negative rational integer, or ¿> . Let Then every number of the form ^ ^ where Vlia a positive rational integer, is a unit of "httff), for y, >*\-l. Also t aucL hence is, a unit of -kCST) • If ^ and ^ are any two distinct positive or negative rational integers, then we have For, s u p p o s e , anu Then <f ± / 19. which is impossible since € > / He&oe every number of the form where >v is a positive or negative rational integer, is a unit of -filiTff) , and any two distinct values of rt give distinct values of e n ; i.e., has an infinite number of units. We have now to show that if Y^  is a unit of •Afa/yJ, then y* tg»> , where n is a positive or negative rational integer or J . Since, if ^ is a unit, a r e "^i^3» w e nee<i consider only the case in which ^ is of the form , where ¿L>0, + Then / and either or That is, 7) ' - f t ijr^e. en> Let JjL-r J » a unit, since Y^  and£*'are units. Then, from 8) l - f ^ * ' But the relation 8) cannot hold, since g is the smallest unit y / . Therefore ^  — / , and Yj- . Then - — £ Y = (e*)' i - i P ^ t <f-n' , and . Hence if ^  is a unit of -tilf?) , ) £ w h e r e and is called the fundamental unit of the realm. ao, Two iategers of iiUT")differing only in a unit factor, are said to be associated. The associates of any integer «C of Ai/ff^t are therefore * w h e r e rx is aero, or any posi-tive or negative rational integer. 10. Prime Numbers of ACi/lr). The definition of a prime number of -hliTf) > is identical with that given for -kiiTa) . To determine whether or not ^ is a prime number of "kCyr?) , we proceed as follows: let 3. =. (a -¡"¿atXe +dto) where are rational integers and cO = . t > A Then Therefore, either ~z - a?+ 9} fr: or 10) I Only 10) has solutions. Hence is a prime. 11. The Unique Factorization Theorem for ¿¿¡J . The Theorems stated in Part I, Art. 10, for hold for ¿SiT)when we replace by jrt^ejj , where «C is any integer of . 21. 12. Rational Prim» Factors of Soma of Integers of -Aliir 1 Theorem. A rational Integer which may be represented by the norm of an Integer of the realm -k(j!r)% i.e., by the form ¿L*'-+(!L4-4r\ has its rational prime factors either (a) primes of the realm -Jk(*l~f)% or (b) norms of primes of the realm. In case (a), these primes enter to even powers, and in case (b), the primes are themselves represented by the form The proof is identical with that given for the correspon-ding theorem for A (¡Pa), In Part I, Art, 11. 13, Representation by the Binary quadratic Form Lemma; The norm of a prime of A V r ) , not associated with a rational prime, is either f , or a rational prime of the form d~n ~t /. The norm of a prime of ALiTr) which is associ-ated with a rational prime is the square of the rational prime, and is of the form J~n ± / . Rational primes of the forms S~n ± a are primes of ACS?) . Every rational prime of the form fn ± / is factorable into two conjugate primes of AUT) and so is the norm of a prime of A¿i/sJm We notice that n[±£ni*-t- > where €=•' and >i is any positive or negative rational integer or zero. Hence and — ^  ~ can each be factored into two conjugate primes of Jtiifir) , and so are not primes of Alt/7') t but are the norms of primes of Let JTts. cl+-4be a prime of Aliir) , which is not as-sociated with a rational prime, and consider the case in W FV"V »(-«Vf "S- . VS" ' * Then h W s aï-feL-4— ¿a. say, .J « , » * is a unique rational prime, since every prime of -hlJf) divides one, and only one, rational prime. But^^^*", 4 una so we have 11) 12) 13} 14) f -- ty » or t % , >ruv6iT. Therefore ^Sh CL — / ynafit^ or a,3--*-<*-4 -! " or a? x ~ or The congruences 11) and 12) have solutions, but 13) «nd I-±) have no solution. Sence altfJ is a rational ^rime o. OX uiLti form £ / . let ^ be a prime of -flU & ) t whieh is aosocia'ced with a rational prime, ^  , say. Then fT-iC*^ , 'where <f=r /-dL^ Z^  and "VLis any positive or negative rational integer or aero. Hence yi£irJ=±^z, and we have >h*A6 National primes of the forms iftii are primes of •A(tTfi)t for a rational prime is factorable into two conjugate primes of -¿¿«/TJoiily if it is / , or of the form A i / . , It remains to show that all prices of the forms fit-1 are factorable into two conjugate primes of In proof, if j> s ±i > > * t then the congruence :¿tood^ has solutions, since ¿"is a quadratic residue of all prices of the forms * * —! . Let 61 be a root . Then , and (a+i/*X<L-t/7)sio>7U)vCl>. But A-tiff and¿t-Vr are integers of . Therefore, if J* is a prime of 'AhF?) , jo must divide either orii-i/7 . m m f P l 2 3 • If where c is an integer of -ACSrj , then U Hence , since jp and &L are rational integers, an therefore p is a divisor of / , which is impossible. Hence f is not a prime of -AU<>) , but is factorable into two conjugate primes of -kLiff) , and so is the norm of a prime of At-fi) . The following Theorem on representation by the binary quadratic form * is an immediate consequence of the Lemma and the Theorem of Art. IE. Theorem. The binary quadratic form a* ir represents and all rational primes of the forms f n i 1, and all rational integers which are products of primes of these forms and even powers of primes of the forms iTn tz . In the latter case, the primes Sn^tZ divide both n, and ^ . The form *can-not represent rational primes of the forms fn > or any ra-tional integer which contains an odd power of a prime of one of these forms. 14. 'The Diophantine Equations : )£%-x(j ~ / » % To find rational integral values of t and y which sat-isfy the equation ^ ' wiiere i s a positive in-teger, we hav'e to find an integer, cC , of A(ifir), whose norm is ^ . Sut Yl[u 7 = e* J , where £ = , and is any positive or negative rational integer, or 2ero, Let m' | are solutions. Also, u p <XA * ^hen ^ , are so-ot the equation 'xf'-f-Xij — J . All solutions of the equations x , -A > 0 % are obtained by let-ting f^ range over all positive and negative rational integers and 0 • Sinoe no two powers of £ are equal, it follows that the number of solutions of each equation is infinite. Consider, first, the equations X V x Cf -J /, J i) and ii) 7. The norm of an integer, << , of-AlSF) is -t / , if, and only if, «C is a unit. If yiAJ = / , «c- Let a, -t-é,-^^ Then X s l « , , satisfy i) . If » P J s - / Let << g . Then x—±clsl , satisfy ii) .• By letting range over all positive and negative rational integers and ^ , all solutions of i) and ii) are obtained. The equations iii) and it) xz+*f where jy is a positive rational prime, have solutions if, and only if o r ¿ J r u e d (Theorem, «rt. 13). Let eC be an integer of AliJr) such that y\JjiJ Jy , and let (¿^•fttŒ* Then x = , y = are solutions of •i « y " " ' then x = , ar ni), if v î L a.+4Ii±J££-, * * A aô. solutions of iv) . All solutions of iii) and iv) are obtained "by letting range as before. The equations y) x V a ^ - y * ^ m and vi) where >?i is a positive rational inte^ox*, Lave solution ift und only if, " " W * •• «K-f. •?.••••?«> where J>,,/!>*. jJu. a/c distinct ^rimea of the ior^s t>ïi "i. I or 6 — ^ d i s t i n c t primea of Lîu for^w f ti , and £ jCi,ttJ s^v^'c positive rational Lu.le-gers. (Theorem, .^a't. 13) . Then if we nave 15) / I3y interchanging one or more of tha /TÍpvíuL their conjuntes in ID), v;e ubtain all possible factorii-ations of m iü^ t. o conjúgate factors. ~e u "771 zo¿. A*. be the result oí c ^ ot.ch jLi-•«/ J a/*/ üerchange, and let r úL^ *^ , 7 » aiU =r CL^-^J ^  CU' . 'hen á; A, t solulions of v' ^ a are solutions of vi), j^ ll soluciona uf v) au! vi) >.rc tue¿_ obtained by letting * o r ^ ^ posiuive ox n^galrve tional integer. tt PAST III THE REALM 4t(fZr») 1. The Numbers of the Realm -AlJ^Ji*)* The number is defined by the equation 1} Since — , every number of -HlJ^Ss)!^ of the form where , } , are rational numbers. Rationalising the denominator, we see that every number of *klf=a$)is of the form 0.-+4 iTas, where Ct and are rational numbers. The other root of 1) defines the realm Jti— tf-as) , conju-gate to -Mf&s), and identical with it. S. The Conjugate ana ITorm of a Kumber of -hCJ^JF*) » If ¥ s (L-f-4rf5a3 is a number of -AtiT^ a), then ¥ =. CL-'ir11-33 is its conjugate. The conjugate of a product of two or more numbers of I-33) is equal to the product of the conjugates of its fac-tors. The norm of is yi [*<£?=. The norm of a product of two or more numbers of % 1 " - • " - 27. j la equal to the product of the norms of its factors. The norm i i* 7 - i I of ere^T number of -HIT**)is evidently a positive rational number. 3. Primitive and Imprimitive numbers of -kti-23) • Every number Y of if-as) satisfies the equation a) x+a.z-+33-4*^.6. If equation S) is irreducible, the number ^ is said to be a primitive number of ftij-23) , and if equation 2) is reducible, y is said to be an imprimitive number of -hC/-¿t3) . The im-primitive numbers of -flll-a?)are the rational numbers. 4. Integers of the Healm •Aif^ad) . Of the imprimitive numbers of All-33) only the rational integers are integers of the realm. If y is an integer of -Atf-as) it may be shorn, as in •AiiTf), that y is of the form 3) y = v SL where ¿t and & are both odd or both even. 5. Basis of By a method similar to that used in -ft¿¡J ¿r), it may be proven that / , are a basis of 4th!-*?) , since every integer of -61 il-tt) can be put in the form <t -t , where ¿t and & are rational integers. If ti>, and are a basis of a theorem identical with that stated for holds for -Aiti-ifd). «ware- v " r»'-' 28. o f WiTW)*. discriminant of -fill-¿3) is / I ~ a 7» The Conjugate and lorm of an Integer of Let t and CO where a) / - De a "basis of tf^as) . < Then 1 and a a > u > ' = . ( > * Therefore ¿¿> satisfies the equation X*— x+6=0. If CL-t^cO is an integer of -ftCiT^dJ, then if'=. a -ftoo'zs. a* +4>h-u>)j and yjJT^ rJ - -t 4u>){ou -r-^ U-tO )} 4) Ylttt = + a rational integer, since a. and are rational integers. 8. Divisibility of Integers of -fiLJ^Zs). An integer «< of -Ar^i/^Jis said to he divisible by an integer f3 of -toItl-as) if there exists an integer Y of Atif^air) such thatdCs^y . 9. Units and Associated Integers. To determine the units of , let £ a unit of -itiiPa3) . Then Hence o^CL- ± Z » and £ -± I . •¡ST" 29. The units of are therefore • + - / , — / • Two integers of -ki/^^S) whioh differ only in a unit factor, are said to be associated. The associates of any in-teger y of -ft(it-3$)are, therefore, + Y , and — Y . 10. Prime numbers of •¿//os}. An integer of fti^-a?) , that is not a unit, and that has no divisors other than its associates and the units, is called a prime number of -ftCxf -33) . In illustration, to determine whether or not J is a prime of we proceed as follows: Let J^rtiOpEM., where X and^y are both odd or Tooth even, and U. and Vare both odd or both even. Then Jj -¿¿±£±#1. tdSa? V* 7 M 4 We have therefore, either 5) or 6) Hi* I -H Prom 5), = if and 2 9 = Neither of these equations have solutions, which are rational integers. Prom 6), and jy = • The solutions of these equations a r e » X =• , o » ^ = ^ . Hence ^ is a prime number of . Similarly, •Z-»4 t/-2 3 may be shown to be a prime number of ~fi(U--33) r 30. 11. The Unique Factorization Theorem in The three theorems upon which the proof of the Unique Factorization Theorem depended in flltTr), do not al-ways hold in -ft Ii£si3) . For l-Jf), Theorem A of M t t ) and * ¿/TV would bu ^ follows: If dC is an integer of -AiiT^sJ and ft it, auj integer of -&C)l-SLl) different from 0 , there exists au j'iilegezytu of such that yi J* J. But if o? /-y- iTJS and fizn S , ket — X -f-Jr^ ä? be the required integer of . •5fr ^  = <r-z -z-)^ For all integral values of ^  , including ¿J , Lhe term23(-5'%)*' is itself > / . Hence / , und til* Z>«'ß]> + That is, v«e cannot find an integer ^¿c such that, anJß'Sj ViLfiJ' Theorem B for -¿¿if-¿3) would "be : If 0C and ^  are any two i l,. U 0 „ r o f inc. t. j^eh other, there exist two Litters ^ and of ni^ J, Le t =r 3 and + , ^ f. lu t ^  = ^ y^JLt&HL . appose ^ ^  3x-t au-zsv— a. <v -f. ¿c -m ir zs.0. 31. Multiplying equation 8} by E, and subtracting the result from 7} we obtain 9) -¿lir- a-Hie left member of 9) is divisible by 3 and the right member only by 2. Hence equation 9} has no solutions in rational integers. That is, we cannot find integers f and of -ACif-SL?) such that, for oCs 3 , and (3 — ft J Theorem 0 for -£¿0-23) would be: If the product of two integers, cC and (3 of -ftill is divisible by a prime number 7T , at least one of the integers is divisible by ff . But ¿a+iFZsXa-iTZa's) t and it may be shown that 3 , H + i/~a 3 , 2— J a r e all primes of , Also the factors of one product are not associated with the factors of the other. Thus,2.*) can be represented in two ways as the product of prime factors, and we see that the Unique Factor-ization Theorem does not hold for -AU -a3) . By the introduction of ideal numbers into ¿V-3d) t the Unique Factorization Theorem may be restored for the realm ) t when factorization is expressed in terms of prime ideal factors, 12, Ideals of . Let , <^be integers of -fc(J^ iFi) . Then the »t. infinite system of integers of the form Si , where each is an integer of -Atrjj) is called an ideal of and we write D\ = (4,,* • 32. The integers «¿„«S, are said to define the ideal , and i s called the symbol of the ideal H The numbers of the infinite system of integers ^ which constitutes the ideal P\ are called the numbers of the ideal. Any integer of -fitf^) which is a linear combination cf the numbers in the symbol of A , may be introduced into the symbol of £N , and any number in the symbol which is linear combination of the remaining numbers in the symbol, may be omitted from the symbol. If yi ss / , £7\ = ¿4,), and is called b j.>rinoip«,l ideal, ¿ill numbers of Eh are then of tha form Y^d, , wnere y is an in-teger of If /Z7y =s. (JL0*L*}—a,, ideal of -Si I it-a'*) , the ideal ES, = whose numbers are the conjugates of the num-bers of • Eh , is called the conjugate of . 13, Equality. Ilultipllcation uii Jiviaion oi Ideals of jkh/^ Zz}. Two ideals, pi = C*,,*», --Vj, h « of -Ahtt) are said to be equal, .aid we write EA b , if every number of A is a number of ¡j , and every number of ¿3 is a numbei' of ; i.e., if and where and are integers of -kiJ-^) . 33 If A*) ^ h =* ({2„ _ are two ideals of At/Z&i) , the ideal defined by all possible products "V f of a number of P\ by a number of ¿3 is called the product of A attd fa • That is, CK h - —Aff'ttf,,' - —A»?'' An ideal of -AliF*?) ia said to be divisible by an i-deal ¿7 of -klf**) if there exists an ideal C of JilJs), such that Ch » b e • Theorem» If an ideal V\ =x. of -hlfw) is divi-sible by an ideal t] - tf.fti,- "klf**) then all numbers of ON belong to tj • Let Ui^bC, where C sr (j[t, \x,--is an ideal of . Then fy ty). But the numbers , '/V'V* Pi a r e numbers of ¿J . Hence all numbers of are numbers of fa . 14. The Unit Ideal. Prime Ideals. Let M = . Then U , r ~ Hence every ideal CK is divisible by the'ideal (1). Let d = Hi, J*, J¡») be an ideal of -id) which di-vides every ideal of Then d¡ divides Let / — d m where m =• — • Then wnere U) = (/,, ^ - - ) . is an integer of V c*f where is an integer of M f T i ) . Henoe / is a number of // * therefore d ^ = Q) . Henoe the ideal (J.) is the only ideal which divides every ideal of kU-2*) . The ideal {1) is therefore called the unit ideal of -AU-a'i) . it contains every integer of -¿M"**) , Ail ideal of J , not the unit ideal, and divisib-le only by itself and the unit ideal, is called a prime ideal of -Alt!-**) . Two ideals of ftU-A*) are said to be prime to each other, or relatively prime, if they have 110 common divisor ex-cept (l). Two integers «< and j3 of •kit"*9) are said to be prime to each other if the principal ideals CoO and (p) are prime to each other, 15, The Unique Factorisation Theorem for -k0f-2$) in Terms of Ideal Factors, The proof of the Unique Factorization Theorem for the ideals of the realm depends upon several other theorems relating to the ideals of the realm. The proofs of these theo-rems and of the Unique Factorization Theorem itself, for the general quadratic realm, are given in 3eid, "The Elements of the Theory of Algebraic numbers", Eence only a statement of the theorems will be given. Theorem 1. There exist in every ideal H of a quadratic realm, two numbers, \t , , such that every number of the i-deal oan be expressed in the form 35. where and are rational integers. Theorem S« in ideal J is divisible by only a finite number of distinct ideals» Theorem 3. If the coefficients U, , , , fl^ of the two rational integral functions of * , + and <yfi)*l&X4/tI are integers of -At/™) and to , an integer of AlJ'™) , di-vides each of the coefficients , , ^ , of the product of the two functions, Ft*)» #(*).(#*) ^ fi f^/J, then each of the numbers <£>(*$, ft*. divisiole by lO Theorem 4. For every ideal £71 of a quadratic realm there ex-ists an ideal b of the realm such that the product Di b is a principal ideal. Theorem 5. If C\ y Jj and C are ideals ancf h = b C , then a-b . Theorem 6. If all numbers of an ideal c belong to an ideal C is divisible by U\ . Theorem 7. If Zj\ and b are any two ideals prime to each other, there exists a number << of Ch «nd a number p o lb such that ©C + fi 1 • Theorem 8. If the product of two ideals, Pi and b , di-visible by a priue ideal p , at least one of the ideals is divisible by p . Cor. 1 p r o d u c t of any number of ideals is divisible by a prime ideal p , at least one of the ideals is divisible " 'W i» * by p • Cor* a . If neither of two ideals is divisible by u crime ideal p , their product is not divisible by p , Cor. 3. If the product of two ideals D\ ¿iid b , is di-visible by an ideal J , and neither JJ\ nor Jj is divisible by J , then J is a composite ideal. The Unique Factorization Theorem for Ideals is proven Ly use of the above theorems. This fundamental theorem reads: Svery ideal can be represented.in une, and onl* one, wc.y «..s oIjb product of prime ideals. The Unique Factorization Theorem enables ua bo dev<=loA. tho arithmetic of ideals» in a maimer anula^ou^ to the devel-opment of the arithmetic of integer a realms i i w^ iioL. fac-torization is unique, n bod^ cf theorem» x^^dil^ u^  ;iveu., but as these are given in Reid for the general quadratic realm, they will not be repeated here. 7/e recall that oh page 31 we indicated the possibility of, more than one factorization lulu jxime;» ux "oho integer A j jLx the realm -AM'**) . .«e close the paper "oj ohov 1liw iii^ u the principal ideal (jf) admits ou.e, and ui.l^  oxie, i*..otor-ization into prime ideals, .«e have - ¿3.) = (* 4 ' 3ut; IS) & (3, ¿ d J ^ / 3 , 37. where f3 , and (3, are prime ideals of m). Also and and ft U* ixLä That is,¿pyjcan Jt factored me, .nl onlj tue, u-^  i^ into ^rimö ide<-l f^etoza. 

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