333 REPORT — 1875. 



where, wlieu the determinant is even, the forms in the second line are 

 always improperly primitive foi-ms. 



Art. 7. [F. 17. Complex Theories.'] 



51. The theory of binary quadratic forms {a, h, c), with complex coefficients 

 of the form a+/3t (j= V — 1 as nsual, <x and /> integers), has been studied by 

 Lejeuuc Dirichlet, Prof. H. J. S. Smith, and possibly others ; but no tables 

 have, it is believed, been calculated. The calculations would be laborious ; but 

 tables of a small extent only would be a suiBcient illustration of the theory, 

 and would, it is thought, be of great interest. 



The theory of complex numbers of the last-mentioned form o+/3i, or say of 

 the niimbers formed with the fourth root of unity, had previously been studied 

 by Gauss ; and the theory of the numbers formed with the cube roots of unity 

 (a+/3w, w" + w + l = 0, o and /j integers) was studied by Eisenstein ; but 

 the general theory of the numbers involving the nth roots of unity (ii an odd 

 prime) was first studied by Ivummer. It will be sufiicient to refer to his 

 memoir, 



Kummer, " Zur Theorie der eomplexen Zahlen," Berl. Monatsb. March 

 1845 ; and Crelle, t. xxxv. (1847), pp. 319-326 ; also " Ueber die Zer- 

 legung der aus Wurzeln der Einheit gebildeten eomplexen Zahlen in 

 ihro Primfactoren," same volume, pp. 327-367, 

 where the astonishing theory of " Ideal Complex Numbers " is established. 



52. It may be recalled that, p being an odd prime, and p denoting a root of 

 the equation p''"^-|-pp~-. . . .-|-p + l=0; then the numbers in question are those 

 of the form « + &p — + /I'p" "', where {a,h ....h) are integers ; or (wh at is in one 

 point of view more, and in another less, general) if tj, r;, . . . . »;,_i are " periods" 

 composed with the powers of p {e any factor of p — 1), then the form con- 

 sidered is ar]-\-h)j^ .... +7(;;^_i. For any value of p or e there is a corre- 

 sponding complex theory. A number (real or complex) is in the complex 

 theory prime or composite, according as it does not, or does, break up into 

 factors of the form under consideration. For p a prime niimber under 23, 

 if in the complex theory N is a prime, then any power of N (to fix the ideas 

 say W) has no other factors than N or N" ; but if p=23 (and similarly for 

 higher values of p), then N may be such that, for instance, W has complex 

 factors other than N or W (for ^5 = 23, N=47 is the first value of N, viz. 

 AT has factors other than 47 and 47") ; say N^ has a complex prime factor A, 

 or we have ^ A as an ideal complex factor of N. Observe that by hypo- 

 thesis N is not a perfect cube, viz. there is no complex number whose cube 

 is = A. In the foregoing general statement, made by way of illustration only, 

 all reference to the complex factors of unity is purposely omitted, and the state- 

 ment must be understood as being subject to correction on this account. 



"What precedes is by way of introduction to the account of Reuschle's 

 Tables (Berliner Monatsberichte, 1859-60), which give in the different com- 

 plex theories p = 5, 7, 11, 13, 17, 19, 23, 29 the complex factors of the 

 decomposable real primes up to in some cases 1000. 



It should be remarked that the form of a prime factor is to a certain ex- 

 tent indeterminate, as the factor can without injury be modified by aftecting 

 it with a complex factor of unity ; but in the tables the choice of the repre- 

 sentative form is made according to definite rules, which are fully explained, 

 and which need not be here referred to. 



53. The following synopsis is convenient : — 



