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- W2009691844 abstract "Letp_ q _1 (mod 4) be distinct primes such that (p|q)=1, 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 (pjq)4 and (qjp)4. In [3], K. Burde proved the following reciprocity law for fourth powers; in this paper, p and q are distinct odd primes, (plq) is the Legendre symbol and (p|q)4= 1 or -1 according as p is or is not a fourth-power residue of q. LEMMA 1. Write p=x2+x2 and q=a2+b2 with xl and a odd, X1X2>0, ab>O and (plq) = 1. Then (p I q)4(q I p)4 = (-1)(-1)14(ax2 bxl q). 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)4( p)4 = or(-l)8, according as q1 or 5 (mod 8), where p=r2+qs2 (see [1]). In the case g= [2, 2, (q+ 1)/2] and q1 (mod 8), Lemma 1 becomes (p Iq)4(q jp)4 = (e| q), where q=2e2_f2 (see [2]). The aim of this paper is to generalize the results of [11 and [2] in the following manner. THEOREM 1. Let p-1 (mod 4) and q_ 1 (mod 8) be distinct primes for which p=kr2+2mrs+ns2, where s is odd and the integralform [k, 2m, n] has determinant q. Suppose each prime divisor of k is a quadratic residue of q. Suppose q=ke2_f 2 for some integers e andf. Then (p I q)4(q | p)4 = (e I q). Proof of this theorem employs the techniques of [2]. First we obtain Received by the editors May 24, 1972. A MS (MOS) subject classifications (1970). Primary 10A15; Secondary 10B05, 10C05. ? American Mathematical Society 1973" @default.
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- W2009691844 date "1973-02-01" @default.
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- W2009691844 title "Biquadratic reciprocity laws" @default.
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- W2009691844 doi "https://doi.org/10.1090/s0002-9939-1973-0313172-8" @default.
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