124 REPORT— 1860. 



systems of ^— 1 units may be established by means of a general proposition 

 due to Dirichlet and relating to any irreducible equation having unity for 

 its first coefficient, and all its coefficients integral. If, in such an equation, 

 R be the number of real, and 2 I of imaginary roots, there always exist 

 systems of R + I — 1 fundamental units, by means of which all other units 

 can be expressed; or, in other words, the indeterminate equation "Norm 

 = 1" is always resoluble in an infinite number of ways, and all its solutions 

 can be expressed by means of R + I— 1 fundamental solutions*. The demon- 

 stration of the actuai existence, in every case, of these systems of fundamental 

 units (a theorem which is, as Jacobi has said f, " un des plus important?, 

 mais aussi un des plus epineux de la science des nombres") is of essential im- 

 portance in the theory of complex numbers, and has the same relation to 

 that theory which the solution of the Pellian equation a? — Dy- = l has to 

 the theory of quadratic forms of determinant D. It may be observed, how- 

 ever, that in the case which we have to consider here, that of the equation 



a x — 1 



t-=0, the existence of fundamental systems of ft — 1 units has been 



ct — 1 



demonstrated independently of Dirichlet's general theory by MM. Kro- 



necker and Kummer %. 



If \=5, a + a _1 is the only fundamental unit; so that every unit is in- 

 cluded in the formula 



If \=7, the complex units are included in the formula 



±aV+*- 1 )"V+a- 2 ) n '- 



But for higher primes the actual calculation of a system of fundamental units 

 involves great labour ; and a method practically available for the purpose 

 has not yet been given. It is remarkable that every unit can be rendered 

 real (*. e. a function of the binary sums or periods a. x -\-a~ 1 , &c.) by multi- 

 plying it by a properly assumed power of a. We shall therefore suppose, in 



* To enunciate Dirichlet's theorem with precision, let/(x) = be the proposed equation ; 

 let «], « 2 , . . . *,, be its roots, and i£(«i), 'r'M* • • ■ *K«») a system of n conjugate units. If 

 the analytical modulus of every one of the quantities ^(*i)' vi/(« 2 ), . . . </<(*,,) be unity, the 

 system of units is an isolated or singular system. The number of singular systems (if any 

 such exist) is always finite, whence it is easy to infer that the units they comprise are 

 simply roots of unity. For if i^(«) be a singular unit, its powers are evidently also singular 

 units, and therefore cannot be all different from one another; i. e. 4>{ct.) is a root of unity. 

 If f(x) be of an uneven order, there are no singular units; if/(.r) be of an even order, —1 

 is a singular unit; and if f(x) = have any real roots, it is the only singular unit; whereas 

 if all the roots of /(a-) = be imaginary, other singular units may iu special cases exist. 



x*~— 1 

 Thus the equation — — r = has 2(\ — 1) singular units included in the formula +«*. Ad- 

 mitting this definition of siugular units, -we may enunciate Dirichlet's theorem as follows : — 

 a system of A units [A = I+R— 1], e^a), e 2 (a), . . . e*(«)> composed with any root «, can 

 always be assigned such that every unit composed with the same root can be represented 

 (and in one way only) by the formula 



« • 'i («)"' • <*(«)"»• e.M" 3 - • • • «*(«)"*. 

 where n,, « 2 , • • • n h are positive or negative integral numbers and w is unity, or some one of 

 the singular units composed with «. 



The principles on which the demonstration of this theorem depends are very briefly indi- 

 cated in the notes presented by Dirichlet to the Berlin Academy in 1841, 18-12, and 1846. 



t Crelle's Journal, vol. xl. p. 312. 



X See Kronecker, De uuitatibus complexis, pars altera ; and Kummer, in Liouville's Jour- 

 nal, vol. xvi. p. 383. 



