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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  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.  Sfa •  Table of Contents, page Introduction,  I  Part I  The Realm M f ^ a ) .  1  Part II  The Realm - H i n r ) .  14  Part III  The Realm -Hit S3).  26  Mtt)*  w  -k(fT) M D  MiT=*3)  Introduction  in algebraic number is a number which satisfies a rational equation, via,, an equation of the form 1)  xn-+a.,xn~+  where Cb, >CLjl)  ==• 0.  ^¿^are rational numbers,  In algebraic integer is an algebraic number which satisfies 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, excluding 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  gfihf  If oC is a number of the second degree, then - 4 6 0  li said to be a realm of the second degree. < / yeaaining root, <<  » of the equation defining «C , is called  the conjugate of cC » a-ad of -/iU) *  I f th  In this case, the  is called the conjugate realm  ® 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. realms -fciO-¿J and  . Vie find, for example, that the -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. in the realm  V/e show that  ~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 rational prime of the form fit 2 / .  or a  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 in which factorization is not unique.  -^YV  *  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  where  9  every number of ^¿jPS)»  are rational numbers.  "the form  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  contains all the numbers of -fcC-i)^  •  -flLJ-S)  contains ail  the numbers of -k(\l-9i) • The Conjugate end L'orm of a number of -fe^i/ -a), If^CsrtfWi^is a number o:C -^¿tPa), the number obt^ineu. from <C by replacing yp^hj iod conjugate-/^ is called tho conjugate of cC If «C =  •  -f-4,and  i ^ a r e 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 ¿positive rational number. If «C an<^  are  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 If  A +-4is  -2.).  a number of -AiyT-z) then  satisfies  the equation a)  y^-zax  whose other root is oC  ¿2-—4r  oy »  Thus every muuber of  P a ) satisfies an equation of the second degree, ber aC is said to be a primitive number of  The num-  ^ 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  3.  from its conjugate.  The imprimitive numbers of  are -fit\f-a)  therefore tlie rational numbers. 4.  Integers of -¿¿.J^g).  A rational number is an algebraic integer when, and only when, it is a rational integer. numbers of A U o n l y  Hence, of the imprimitive  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 rational integers. Let  a^-^/SbQ aii integer of -AitiZfi) •  also aii integer of -A(ii-s) .  Then cC /-il  The rank e^uaiions of <<  are  3)  *z-aa.,x  4, *= o,  and a Oj  4) respectively.  Therefore, from Z>) c*nd 4), Sd,  *77t//  a  ^a«ionai integer;  and 4 Hence, we have 5) i J in 1 !."), „I t Substituting 5}  si >vf  u rational L^t^er.  is and «fi/s?  t  Therefore 2ZL-#%-*."' V ' ' JLf  V, is a rational integer, and we have «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 t  Therefore, if  At Af  of A(J~Hr)>  form  i* au integer  -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 expressed 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. are a basis of •ACif^ii) •  Hence / and  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 Hence /-f tf-a and Theorem.  If  a rational integer.  are a basis of and  are a basis of JtiJ^a) the necessary  and sufficient condition that  are  where of  is  integers, shall be also a basis  a, a^ l-tl-  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  k j ' ^ t ^ i is divisible by f-il-2 since  J+JiT:a 8*  (l-f^X-i+F*).  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. c  /  where «c is an integer of  Hence  )•  . • X=* ± The units of  (f  ) 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 n o t 5 is a prime number of -ACiP*) we proceed as follows: Let  \$  iFaXc  ), where  integers.  Then either 51  r ^ ^ l , Lc*  e,d. are rational  ^  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) Elements of the Theory of Algebraic numbers".  in Reid, "The 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 such that  and y  of  J-5)  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 g  , and there exist two integers  and ^ of ~k(f-a) such that Gor. 2,  If <tt)4*f  ,*tnare any >v integers of  if 1 *)  there exists a common divisor J" of common divisor  divides  -vJ.such that every , and there exist  yu in-  tegers  -  jJ n such that  Theorem C.  If the product of two integers < and (3 of ^CtT^i)  is divisible "by a prime number V gers is divisible by W Gor, 1,  , at least one of the inte-  .  If the product of any number of integers of -k(itt)  is divisible by a prime number (T > at least one of the integers 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. to  is prime  ar .  Cor. 3. y  If«< an* fl are each prime to y , then  , ^  11.  If«(, is prime to Y , and if <£(3 is divisible by is divisible by  .  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,,^, distinct.  are primes of the realm, not necessarily  Then Vi£Vr/  is a rational integer, and one of the  type specified in the Theorem. Let  n[rr]~ fAff'. . . •  where fajpt> — ?, tZf  - - are distinct rational primes, and 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. 8} where  Let fr^SK*  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 prime of A ( \ J •  is a unit, and hence that  Since  yxfrjz nfaJ nEfrJ' • • jot occurs to an even power in IS.  7 .•  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 8 n + / • Rational primes of the forms 8h—I and 8/1 prime of the form 8n+l  are primes of -¿¿if1.2) • or \$n+\$  Every rational  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  yiM  fT  ¿"/-a  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 • hare yiQrJs/» t W ^  Then /T= ¿y, , and ntnls.and  •  Rational primes of the forms 9n~t primes of  we  and  are  , 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, i f ^ S / or3 >7*4,9 , then the congruence has solutions, since  is a quadratic residue of all primes  of the forms 8r\ + 1 and 9n + 3 * Let a, be a root. £tZ~  ryuUt^  Then  >n*dfi, and  ((X -+ But ¿L-t-1/-Si and ¿2 — / - a if yb is a prime of CL-lP^i •  . are integers of -fcCO  Therefore,  , yb must divide either ¿L-t-if-k or  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 representation by the binary quadratic form Theorem.  ¿¿Va^*.  z  /a  The binary quadratic form CL 't&'v , represents «?  and all positive rational primes of the forms ^ n y - /  and  Pyl*+3 , and all positive rational integers which are products of primes of these forms, and even powers of primes of the forms  —/  and  Pn-3  primes divide both a. and  %  In the latter case, the  .  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 satisfy these equations, we have to find an integer <C of *AC\J-a) whose norm is the right member of the equations. then X st  *  tf—  are  If dCs/t+^lfi-k  solutions.  Consider, first, the equation i) The integer £  I. satisfies this equation if, and only if,  a unit of -AU^i)*  Hence  as the only solutions of i).  ± J , and we have  /, ^  is &  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, ynfid.8 •  If  or  5 is an integer of -AtiT^)  whose norm is Jo , then Xsr+4, ,  are the solutions of  ii). Consider, finally, the equation iii)  x^sL^ytL,  where >7tis a positive rational integer.  By the Theorem of  Art. 11, equation iii) has solutions if, and only if, ^«i»8^® distinct primes of the forms  where and tn+'Z  ¿L ;  forms  • > d i s t i n c t primes of the  and  rational integers.  +l  , and c,,l\tt —  --  positive  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 m i n t o conjugate factors.  Let  change.  Then  •J «/  V  'Jbe the result of any such inter» 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  !•  Mil)  The lumbers of the Realm 'hi\T6)*  The number /¿T is defined by the equation 1)  -  Since  =r CT , every number of ^ ¿ V ^ i s of the form t  where  are rational numbers.  Ra-  tionalizing the denominator, we see that every number of is of the form  ¿t -h-tiTf, where ¿Z and -¿'are rational numbers. The other root of 1) defines the realm to  . As in -ftCiFii) , the realms  fif)*  conjugate  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 reducible, 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 integers are integers of Let ft-^+J; iff be an integer of -A/VTj . also an integer of -foiiT^) •  Then /^/F 5  The rank equations of ft , and  are  3)  yc - 2afx -HI*-  o,  and  = O,  4) respectively.  Prom 3) and 4), — Vrvf /¿^ =  That is,  5)  a rational integer tt  ft: A rv  Id  Substituting in 3) ,  u  ir  is  X ~~>7t.X therefore  -jfe =5 ¿>  is a rational integer, and we have fvnz-nxs.6  yrmJl-y  ' . n ^ o y n t d f and Let  Then  vyi*-  since f&jyruML/j •  /j ; i.e.,  Therefore  are both odd or both even. Hence every integer of  where <t and  y  yjuwia . Hence >n and -/f  But VI—  4 , and  = jjt>  is of the form  are both odd or both even.  Basis of The integers of AhiT) are of the form and •if'are both odd or both even.  where ¿2,  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  al integer since CL and Hence, every integer  a ration-  are both odd or both even. of -Alif?) can be expressed in  the form j-tjS* where X a n d ^ are rational integers.  Therefore  / and /  17.  are a basis of At\TF) • A theorem similar to that stated in Part I, Art. 5, for rA/tT5) holds for ALifT).  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  Then  (O ss. y^jTW f he a basis of «z » ancl eO+cfi'-z / , ¿^¿¿/s—/. Hence  co'~  satisfies the equation  * ' X — / »¿J -  Let ^ss a-f-^o) be an integer of  and  (p.+~4'u>J(&-rM/-a>)}  SL ¿L-t-CL  i  Then  a  -+~4rS<0 —  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 9.  =-¿3-  The Units of -hLiTF)*  iTr) -  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. " !  v - +/±iT=T  which  y - ft * ^ . «2.  which  l  are not integers of -hlt/s); or else  and has the solutions  V =  ±jtlr ^  are integers arid units of Jiti/IF)*  When  »  3etc., other  units of "A(ifF)may be determined in a similar way. .Evidently / i f i r Theorem.  is the smallest unit greater than / •  .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, Also  >*\-l.  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^ y* tg»>  is a unit of •Afa/yJ, then  , where n is a positive or negative rational integer  or J  . are  Since, if ^ is a unit, consider only the case in which ^ where ¿L>0,  Then  "^i^3»  we  nee<i  is of the form  ,  +  /  and either  or  That is,  7)  ' -  Let JjL-r J  »  8)  a  ijr^e. ft  en> unit, since Y^ and£*'are units.  Then, from  l-f^*'  But the relation 8) cannot hold, since g is the smallest unit y  / .  Therefore ^ —  Y = (e*)'  /  , and Yj-  i - i P ^ t <f-n'  .  Then -  , 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 are said to be associated. of Ai/ff^t are therefore  only in a unit factor,  The associates of any integer «C * 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  t >  are rational integers and cO =  A  .  Then  Therefore, either  fr:  ~z - a?+  9} or  I  10)  Only 10) has solutions. 11.  Hence  is a prime.  The Unique Factorization Theorem for ¿¿¡J  .  The Theorems stated in Part I, Art. 10, for hold for  ¿SiT)when we replace  any integer of  .  by jrt^ejj , where «C is  21.  12.  Rational Prim» Factors of S o m a 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 corresponding 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 form d~n ~t /.  f , or a rational prime of the  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 f n ± / is factorable into two conjugate primes of  AUT) and so is the norm of a prime of A¿i/sJm We notice that  n[±£ n i*-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  FV"V »(«Vf "S- . VS" ' *  W  ¿a.  Then  say, h W s aï-feL-4— « , » * is a unique rational prime, since every prime of -hlJf)  .J  divides one, and only one, rational prime. so we have f -- ty  » or t % , >ruv6iT.Therefore  ^Sh CL —  11)  B u t ^ ^ ^ * " , 4una  / ynafit^ or  12)  a,3--*-<*-4  -! "  or  13}  a?  x  or  ~  14) The congruences 11) and 12) have solutions, but 13) «nd I-±) have no solution. form  Sence  altfJ is a rational ^rime o. OX uiLti  £/. 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 are primes of •A(tTfi)t  National primes of the forms iftii  for a rational prime is factorable into two conjugate primes of -¿¿«/TJoiily if it is /  , or of the form  Ai/.,  It remains to show that all prices of the forms fit- 1 are factorable into two conjugate primes of if j> s ±i > > *  t  In proof,  then the congruence : ¿ t o o d ^ 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 is a prime of 'AhF?) , jo  .  Therefore, if J*  must divide either  orii-i/7 .  m m  23  fPl  •  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 . primes Sn^tZ divide both n, and ^  .  In the latter case, the 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 isfy the equation  ' wiiere  ^  is a  which sat-  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.  <XA ot the equation  * ^hen  ^  'xf'-f-Xij —  J  of the equations x  Also,  up  ,  are so-  .  All solutions  , -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  i)  /,  J  and ii)  7.  The norm of an integer, << , of-AlSF) is -t / , if, and only if, «C is a unit.  If yiAJ =  / , «c-  Then X s l « ,  , satisfy i) .  Let << g  .  By letting  Then x—±clsl  Let  a, -t-é,-^^  If » P J s - / ,  satisfy ii) .•  range over all positive and negative rational  integers and ^  , all solutions of i) and ii) are obtained.  The equations iii) and z  it)  x  +*f  where jy is a positive rational prime, have solutions if, and only if  or¿Jrued  (Theorem, «rt. 13).  be an integer of AliJr) such that y\JjiJ Jy  ni),  Let eC  , and let  (¿^•fttŒ* Then x = , y = are solutions of •i « y , ar if v î" L a". + 4 I'i ± J £ £ - , then x =  *  *  A  aô. solutions of iv) . "by letting  All solutions of iii) and iv) are obtained  range as before.  The equations  y)  xVa^-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 f ti gers.  •?.••••?«>  —  , and £  ^distinct  jCi,ttJ  primea of Lîu for^w  s^v^'c positive rational Lu.le-  (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 i ü ^ t. o conjúgate factors.  ~e u "771 zo¿. A*. be the result oí c ^ ot.ch jLi•«/ J a/*/ r úL^*^, 7  üerchange, and let tt 'hen á; A,  t  are solutions of vi),  j^ll soluciona uf v) au! vi) >.rc tue¿_  obtained by letting tional integer.  » aiU  =r CL^-^J ^ CU' .  solulions of v' ^ a  *  or  ^ ^ posiuive ox n^galrve  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) , conjugate 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  % j  1  "  - • "  -  27.  la equal to the product of the norms of its factors.  i i* 7 -  i  The norm  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  x+a.z-+33-4*^.6.  a)  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)  vy =  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. Ifti>,and  are a basis of a theorem identical  with that stated for  holds for -Aiti-ifd).  «ware- v " r»'-'  28. of  WiTW)*.  discriminant of  -fill-¿3) is  / 7»  I ~a  The Conjugate and lorm of an Integer of Let t and CO where  Then  1  a)  / <-  CL-t^cO  a "basis of  and a a > u > ' = . ( > *  satisfies the equation X*— If  De  tf^as) .  Therefore ¿¿>  x+6=0.  is an integer of -ftCiT^dJ, then  if'=. a -ftoo'zs. a* +4>h-u>)j and  4)  yjJT^rJ -  -t 4u>){ou -r-^U-tO )}  Ylttt =  +  a rational integer, since a. and 8.  are rational integers.  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 Atif^air) such thatdCs^y . 9.  Units and Associated Integers. To determine the units of  a unit of -itiiPa3) . Hence  , let £  Then  o^CL- ± Z » and  £  -± I .  Y  of  •¡ST"  29. The units of  are therefore  •+-/,—/  •  Two integers of -ki/^^S) whioh differ only in a unit factor, are said to be associated. teger y of 10.  The associates of any in-  -ft(it-3\$)are, therefore, + Y , and — Y  .  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  We have therefore, either  4  5) or  Hi*  6)  I  H  Prom 5),  =  if  and 2 9 =  Neither of these  equations have solutions, which are rational integers. 6),  and  jy =  these equations a r e »  X =•  Hence ^ is a prime number of  • ,  o  Prom  The solutions 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* But if  ket  o?  J.  /-y- iTJS  and fizn S  ,  — X -f-Jr^ä? be the required integer of •5fr ^  = <r-z  For all integral values of ^ is itself > /  .  .  -z-)^  , including ¿J , Lhe term23(-5'%)*'  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 other, there exist two Litters ^  Le t  =r 3  y^JLt&HL  and .  +  of  , ^f. lu t ^ =  appose  3x-t au-zsv— <v  and  inc. t. j^eh ni^J,  ^ ^^  a.  -f. ¿c -m ir zs.0.  31.  Multiplying equation 8} by E, and subtracting the result from 7} we obtain  9)  - ¿ l i r - a-  Hie left member of 9) is divisible by 3 and the right member only by 2. integers.  Hence equation 9} has no solutions in rational That is, we cannot find integers f  and  -ACif-SL?) such that, for oCs 3 , and (3 —  of J  ft  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 3  .  ¿a+iFZsXa-iTZa's)  , H + i/~a 3 , 2— J a r e  t  and it may be shown that  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 Factorization 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) . »t.  infinite system of integers of the form Si is an integer of -Atrjj) is called an ideal and we write D\ = (4,,*  •  Then the  , where each of  32. are said to define the ideal  The integers «¿„«S, is  , and  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,, ES, =  whose numbers are the conjugates of the num-  bers of • Eh 13,  ideal of -Si I it-a'*) , the ideal  , is called the conjugate of  .  Equality. Ilultipllcation uii Jiviaion oi Ideals of jkh/^Zz}. Two ideals, pi = C*,,*», --Vj, h «  are said to be equal, .aid we write EA of A of  is a number of ¡j  of -Ahtt) b  , if every number  , and every number of ¿3 is a numbei'  ; 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 A  attd fa •  CK h -  That is,  An ideal  is called the product of  —Aff'ttf,,'  - —A»?''  of -AliF*?) i a said to be divisible by an i-  deal ¿7 of -klf**) if there exists an ideal C such that Ch » b e  •  Theorem»  V\ =x.  If an ideal  of JilJs),  of - h l f w ) 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 of  ¿J  14.  ,  '/V'V*  . Hence all numbers of  The Unit Ideal.  Pi  are  numbers  are numbers of fa .  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  vides every ideal of Let / — d m  where m  -id) which di-  Then d¡ divides =•  —  •  Then  U) = (/,,  wnere  V  . is an integer of  ^  --  )  c*f  where //  is an integer of *  MfTi)  .  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 except  (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  realm, two numbers, \t ,  of a quadratic  , such that every number of the i-  deal oan be expressed in the form  35. where  and  Theorem S«  are rational integers.  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 vides each of the coefficients  ,  AlJ'™)  , di-  , ^ , of the product  of the two functions,  Ft*)» #(*).(#*) ^fi then each of the numbers by  <£>(*\$,  f^/J, ft*.  divisiole  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.  .  a-b  Theorem 6. C Theorem 7.  If C\ y Jj and C are ideals ancf  If all numbers of an ideal c belong to an ideal is divisible by U\ If Zj\ and  .  b are any two ideals prime to each  other, there exists a number << such that Theorem 8.  h = b C , then  of Ch «nd a number p  o lb  If the product of two ideals, Pi and b ,  di-  ©C + fi  1  •  visible by a priue ideal p divisible by p  .  , at least one of the ideals is  Cor. 1 p r o d u c t by a prime ideal p " 'W i» *  p  by  , at least one of the ideals is divisible  •  Cor* a . ideal p Cor. 3.  If neither of two ideals is divisible by u crime , their product is not divisible by  p  If the product of two ideals D\ ¿iid  visible by an ideal J by J  of any number of ideals is divisible  , then J  , and neither JJ\  , b  nor Jj  , is di-  is divisible  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 development of the arithmetic of integer a torization is unique,  realms i i w^iioL. fac-  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 jLx the realm  -AM'**) .  lulu  jxime;» ux "oho integer A j  .«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  of  f3 ,  and  (3,  are prime ideals  m).  Also  and  and ft  U*  That is,¿pyjcan Jt factored into ^rimö ide<-l f^etoza.  ixLä m e , .nl onlj tue, u-^ i^  `

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