134 REPORT — 1860. 



It would seem that for certain values of the prime X, there exist classes of 

 ideal numbers appertaining to the exponent H, if H denote the number of 

 classes of ideal numbers*. Such classes (when they exist) possess a property 

 similar to that of the primitive roots of prime numbers ; i.e., by compounding 

 such a class continually with itself we obtain all possible classes, just as by 

 continually multiplying a primitive root by itself we obtain all residues 

 prime to the prime of which it is a primitive root. It has, however, been 

 ascertained by M. Kummer that these •primitive classes do not in all cases, 

 or even in general, exist. 



The theorem of this article enables us to express ideal numbers as roots 

 of actually existing complex numbers. Thus, if q be a prime appertaining 

 to the exponenty for the modulus X, and resoluble into the product of e con- 

 jugate ideal factors <(>(ri ), <p(vi)> "K^)* • • • (fi^e-i)' these ideal numbers, which 

 will not in general belong to the same class, will nevertheless appertain to 

 the same exponent h ; so that \_<p(ji )~\ h , [_<p(ji^)~\ h > • • • will all be actual num- 

 bers. The power q 1 ' is therefore resoluble into the product of e actually 

 existing complex factors. If we effect this resolution, and represent the 



factors of q h by 3>(j? )> *(>7i) • • • • > the ideal numbers 0(>? o ), ^C^), • • • • lnav be 

 represented by the formulae 



i i 



*0».)= [*0»o)] x . *0h)= [*('/!>] ~ h , 



50. The Number of Classes of Ideal Numbers. — The number of classes of 

 ideal numbers was first determined by Dirichlet. He effected this determi- 

 nation by methods which he had previously introduced into the higher 

 arithmetic, and which had already led him to a demonstration of the cele- 

 brated theorem, that every arithmetical progression, the terms of which are 

 prime to their common difference, contains an infinite number of prime 

 numbers, and to the determination of the number of non-equivalent classes 

 of quadratic forms of a given determinant f. Dirichlet's investigation of the 

 problem which we are here considering has never been published ; but that 

 since given by M. Kummer is probably in all essential lespects the same, as 

 it reposes on an extension of the principles developed in Dirichlet's earlier 

 memoirs. Our limits compel us to omit the details of M. Kummer's analysis ; 

 the final result, however, is, that if H denote the number of non-equivalent 



P D 



classes of ideal numbers, H= 7 — X — In this formula P is a quantity 



defined by the equations 



P=0(/3)K/3 3 )K/3 5 ).--0(/3'- 2 ), 



0(/3)=l+y 1 /3 + y 2 /3 2 + 7a /3 3 + ... +y K - 2 P~\ 



* See on this subject M. Kummer's note "on the Irregularity of Determinants " in the 

 Monatsberichte of the Berlin Academy for 1853, p. 194. M. Kummer's investigation, 



however, is restricted to classes containing ideal numbers /(«) such that/(a)x/(« _1 ) is 

 an actual number. 



t See his memoirs on Arithmetical Progressions, in the Transactions of the Berlin Academy 

 for the years 1837 (p. 45) and 1841 (p. 141), or in Liouville, vol. iv. p. 393, ix. p. 255. The 

 first of these papers relates to progressions of real integers, the second to progressions of 

 complex numbers of the form a-\-bi. In the memoir " Hecherches sur diverses applications 

 de l'analyse iunnitesimale a la Theorie des Nombres" (Crelle, vol. xix. p. 24, xxi. pp. 1, 

 & 134), Dirichlet has applied his method to quadratic forms having real and integral co- 

 efficients ; and in a subsequent memoir (Crelle, vol. xxiv. p. 291), he has extended this ap- 

 plication to quadratic forms, of which the coefficients are complex numbers containing i. 

 See also Crelle, vol. xviii. p. 259, xxi. p. 98 (or the Monatsberichte for 1840, p. 49), xxii. p. 

 375 (Monatsberichte for 1841, p. 190). We shall have occasion, in a later part of this Report, 

 to give an abstract of the contents of this inTaluable series of memoirs. 





