105 



On the Decomposition of Prime Numbers in a Biquadratic 

 ^ I^umber-Field. 



By Jacob Westlund. 



Let 



x^4-ax2 + bx + c = 



be an irreducible equation with integral co-eflScients, whose discriminant ^^ we 

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

 and let us consider the number-field k(9), generated by 9. Then since the funda- 

 mental number of k(9) enters as a factor in the discriminant of every algebraic 

 integer in k(9), it follows that /^ is the fundamental number of k(9) and 



1, 9, 92, 93 

 form an integral basis, i. e., every algebraic integer <x in k (9) can be written 



oc = ao + ai9 + 3292 + 3393 

 where 3o, 3i, 32, as are fetional integers. 



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

 effected by means of the following theorem : If 



F(x) = x4 + ax2-f bx + c 

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



F(x)={ Pi(x) }"'[ 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, Pi(9)]'i [p, P2(9)]'2 



where p, Pi (9) , p, P2(9) are different prime ideals of degrees fi, 



f2,... respectively. (1) 



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

 sider, 1st when p ^ ^^ and 2nd when p =fc z^. 



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 (9) is divisible by p'l Ci — ^)+ '2(^2 — ') 

 r • • ' ( )) we have 



fiCi — l) + f2(^2 — 1) +f3(«^3 — ^) + U(U — ^) = 1, 



(1) Hilbert: "Bericht liber die Theorie der Algebraischen Zahlkorper," Jahre»bericht 

 der Deutschen Mathematiker- Vereinigung (1894-95), pp. 198* 202. 



8— A. OF SCIKNCE. 



