252 report — 1859. 



numbers a for which the congruence x A =a, mod p, is resoluble ; that is, the 

 biquadratic residues ofp; the third comprises those numbers which are qua- 

 dratic, but not biquadratic, residues of p ; the second and fourth classes divide 

 equally between them the non-quadratic residues. 



We owe to Gauss two memoirs* on the Theory of Biquadratic Residues, 

 which, while themselves replete with results of great interest, are yet more 

 remarkable for the impulse they have given to the study of arithmetic in a 

 new direction. Gauss found by induction that a law of reciprocity (similar 

 to that of Legendre) exists for biquadratic residues. But he also discovered 

 that, to demonstrate or even to express this law, we must take into con- 

 sideration the imaginary factors of which prime numbers of the form 4/J + l 

 are composed. By thus introducing the conception of imaginary quantity 

 into arithmetic, its domain, as Gauss observes, is indefinitely extended ; nor 

 is this extension an arbitrary addition to the science, but is essential to the 

 comprehension of many phenomena presented by real integral numbers them- 

 selves. 



Gauss's first memoir (besides the elementary theorems on the subject) con- 

 tains a complete investigation of the biquadratic character of the number 2 

 with respect to any prime p=4?n + l. The result arrived at is that if p be 

 resolved into the sum of an even and uneven square (a resolution which is 

 always possible in one way, and one only), so that p=a 2 + b 2 (where we may 

 suppose a and b taken with such signs that «=1, mod 4 ; b=af, mod p), 2 

 belongs to the first, second, third, or fourth class, according as \b is of the 

 form 4h, 4»+l, 4rc-f-2, or 4w + 3. The considerations by which this con- 

 clusion is obtained are founded (see Art. 22 of the memoir) on the theory of 

 the division of the circle, and we shall again have occasion to refer to them. 

 In the second memoir Gauss developes the general theory already referred to, 

 by which the determination of the biquadratic character of any residue of 

 p may in every case be effected. The equation p=a 2 + b 2 shows that p= 

 (a+bi)(a—bi), or that p, being the product of two conjugate imaginary 

 factors, is in a certain sense not a prime number. Gauss was thus led to 

 introduce as modulus instead of p one of its imaginary factors: an innovation 

 which necessitated the construction of an arithmetical theory of complex 

 imaginary numbers of the form A + B/. The elementary principles of this 

 theory are contained in the memoir in question ; they have also been developed 

 by Lejeune Dirichlet with great clearness and simplicity in vol. xxiv. of 

 Crelle's Journal (pp. 295-319, sect. l-9)t. The following is an outline of 

 the definitions and theorems which serve to constitute this new part of arith- 

 metic. 



25. Theory of Complex Numbers. — The product of a number a +bi by its 

 conjugate a — bi is called its norm; so that the norm of a + bi is a 2 +b~ ; the 

 norm of a (which is its own conjugate) is a 2 . This is expressed by writing 

 N(a+fo')=N(a— bi)=a 2 + b 2 ; N(a)=a 2 . If a and ft be two complex num- 



* Theoria Residuorum Biquadraticorum. Commentatio prima et secunda. (Gottingse, 

 1828 and 1832, and in the Comm. Recent. Soc. Gott., vol. vi. p. 27 and vol. vii. p. 89.) The 

 articles in the two memoirs are numbered continuously. The dates of presentation to the 

 Society are April 5, 1825, and April 15, 1831. 



f The death of this eminent geometer in the present year (May 5, 1859) is an irrepa- 

 rable loss to the science of arithmetic. His original investigations have probably contributed 

 more to its advancement than those of any other writer since the time of Gauss ; if, at least, 

 we estimate results rather by their importance than by their number. He has also applied 

 himself (in several of his memoirs) to give an elementary character to arithmetical theories 

 which, as they appear in the work of Gauss, are tedious and obscure ; and he has thus done 

 much to popularize the theory of numbers among mathematicians — a service which it is im. 

 possible to appreciate too highly. 



