105 



On the Decomposition of Prime Numbers in a Biquadratic 

 Number-Field. 



By Jacob Westlund. 



Let 



x^ + ax2 + bx + c = 



be an irreducible equation with integral co-eflBcients, whose discriminant /\ we 

 suppose to be a prime number. Denote the roots of this equation by 9, 0', 0", 0'", 

 and let us consider the number-field k(0), generated by 0. Then since the funda- 

 mental number of k(0) enters as a factor in the discriminant of every algebraic 

 integer in k(0), it follows that ^ is the fundamental number of k(0) and 



1, 0, 02, 03 

 form an integral basis, i. e., every algebraic integer a in k (0) can be written 



a z^ ao -f- ai0 -\- 320^ + 330^^ 

 where ao, ai, 32, as are rational integers. 



The decomposition of any rational prime p into its prime ideal factors is 

 effected by means of the following theorem : If 



FU) = x< + ai2 -1- bi + c 

 be resolved into its prime factors with respect to the modulus p and we have 



F(x)E={ Pi(x) y'[ P2(x) Y' (modp) 



where Pi{x), P2(x). .. are different prime functions with respect to p, of degrees 

 fi, f2, . . . respectively, then 



(p)=fp, Pl(0)]'' [p, P2(0)j'2 



where p, Pi(0) , p, P2(0) are different prime ideals of degrees fi, 



f2,... respectively, (i) 



In applying this theorem to the factorization of p we have two cases to con- 

 sider, 1st when p ^ /^ and 2nd when p =b A. 



Case I. p = ^. 

 Suppose 



(p) == A^' A^= A«^ A^* 



where Ai, A2 . . . are different prime ideals of degrees fi, f2, ..., respectively. 

 Then, since the fundamental number of k (0) is divisible by p 1 C^i )+ 2(*2 ) 

 ~r • • • ( )) we have 



fl^l — l) + f2(^2 — 1) +isi'3'-^) +f4(^4 — i) = l, 



(1) Hilbert: "Bericht iiber die Theorie der Algebraischen Zahlkorper," Jahrefbericht 

 der Deutschen Mathematiker-V^ereinigung (1894-95), pp. 198> 202. 



8— A. OF SCIKNCE. 



