ON THE THEORY OF NUMBERS, 153 



risen of a certain form, of which the resolvent function of any Abelian 

 equation is susceptible, with M. Kummer's expression for the resolvent func- 

 tion in the case of the equation of the division of the circle (see art. 60). 

 It thus involves considerations relating to ideal numbers. 



Two propositions of a more special character, and closely connected with 

 one another, have also been given by M. Kronecker (Crelle, vol. liii. p. 173). 

 Their demonstration is immediately deducible from the principles of Dirich- 

 let's theory of complex units : — 



" If unity be the analytical modulus of every root of an equation, of which 

 the first coefficient is unity and all the coefficients are integral numbers, the 

 roots of the equation are roots of unity." 



"If all the roots of an equation (having its first coefficient unity and all 

 its coefficients integral) be real and inferior in absolute magnitude to 2, so 

 that they can be represented by expressions of the form 2 cos a, 2 cos /3, 

 2 cos y, . . . . the arcs a, /3, y are commensurable with the complete circum- 

 ference." 



In the following proposition M. Kronecker has extended a theorem of 

 M. Kummer's (art. 42) relating to complex units composed with roots of 

 unity of which the index is a prime, to complex units composed with any 

 roots of unity (Crelle, vol. liii. p. 176) : — 



" Every complex unit composed with the roots of the equation w n =],can 

 be rendered real by multiplication with a 4wth root of unity. If n be even, 

 a 2«th root will always suffice ; and if n be a power of a prime, an wth root 

 will suffice." 



The demonstration of this proposition is also deducible from Dirichlet's 

 principles. 



63. Tables of Complex Primes — In M. Kummer's earliest memoir on 

 complex numbers (Liouville, vol. xii. p. 206) he has given a table of the 

 complex factors, composed of Xth roots of unity, which are contained in real 

 primes of the form w\ + l inferior to 1000, X representing one of the primes 

 5, 7, 11, 13, 17, 19, 23. This memoir was written before M. Kummer had 

 considered the complex factors of primes of linear forms other than m\+l, 

 and before he had introduced the conception of ideal numbers. The com- 

 plex prime factors of real primes of those other linear forms are, therefore, 

 not exhibited in the Table; and the five numbers of the form 22m +1, 47, 

 139, 277, 461, 967, each of which contains 22 ideal factors composed of 23rd 

 roots of unity, are represented as products of 11 actual factors (each of 

 which contains two reciprocal ideal factors). The tentative methods by 

 which the complex factors were discovered are explained in sect. 9 of the 

 memoir cited. Since the full development of M. Kummer's theory, Dr. 

 Reuschle has undertaken to complete and extend the Table. He has already 

 given tables containing the complex prime factors of all real primes less than 

 J000, composed of 5th, 7th, 11th, 13th, 17th, 23rd, and 29th roots of unity, 

 together with the complete solution of the congruences corresponding to the 

 equations of the periods (see the Monatsberichte for 1859, pp. 488 and 

 694, and for 1860, pp. 150 and 714). For 5, 7, 11, 13, 17, the complex 

 primes are exhibited in a primary form ; for 19, 23, and 29 they are exhibited 

 in a form which satisfies the condition /(a) =/(!), mod (1— a) 2 , but not 

 the condition / (a) /(«-!) = [/(I)] 2 , mod X. The ideal factors Dr. 

 Reuschle represents by their lowest actual powers ; for 23 this power is the 

 cube, for 29 it is the square; for 1 1, 13, 17, 19, as well as for 5 and 7, all 

 complex prime factors of real primes less than 1000 are actual. It appears 

 from the Table (and it has indeed been proved by M. Kummer), that 29 is 

 an "irregular determinant" (see art. 49, note) ; for the number of classes is 



