PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 37, Number 2, February 1973 BIQUADRATIC RECIPROCITY LAWS EZRA BROWN Abstract. Let p=q=\ (mod 4) be distinct primes such that (p\q)=l, and let g = [k,2m, n] be a binary quadratic form of determinant q which represents p. Subject to certain restrictions on k and q, we obtain some reciprocity laws for the fourth-power residue symbols (p\q)¡ and (q\p\. In [3], K. Bürde proved the following reciprocity law for fourth powers; in this paper,/? anda are distinct odd primes, (p\q) is the Legendre symbol and (p\q)n=\ or —1 according as p is or is not a fourth-power residue of a. Lemma 1. Writep=x\+x\ andq=a2+b2 with x, and a odd, jc,x2>0, ab>Oand(p\q)=l. Then (P I qUq | P\ = (-l)"-1)/4(ax2 - bxx | a). This result can be formulated in terms of a representation of p by a form g of determinant q. In the case g= [1, 0, q], Lemma 1 has the form (p\q)Áq\p)4 = I or (-iy, according as q=\ or 5 (mod 8), where p=r2+qs2 (see [1]). In the case g—[2,2, (a+l)/2] and q=\ (mod 8), Lemma 1 becomes (p\q)i(q\p)i = (e\q)> where q=2e2—f2 (see [2]). The aim of this paper is to generalize the results of [1] and [2] in the following manner. Theorem 1. Let p= 1 (mod 4) andq= 1 (mod 8) be distinct primes for which p=kr2+2mrs+ns2, where s is odd and the integral form [k, 2m, n] has determinant a. Suppose each prime divisor ofk is a quadratic residue of q. Suppose q=ke2—f2 for some integers e andf. Then (pI q\(q I p)* = (e I q)- Proof of this theorem employs the techniques of [2]. First we obtain Received by the editors May 24, 1972. AMS (MOS) subject classifications (1970). Primary 10A15 ; Secondary I0B05,10C05. © American Mathematical Society 1973 374 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use