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Residuacity of Primes
Abstract
Let q,p be distinct primes with p - ef + 1. A variant of the Kummer-Dedekind theorem is proved for Gaussian periods, which shows in particular that q is an e-th power (modp) if and only if the Gaussian period polynomial of degree e has e (not necessarily distinct) linear factors (modg). This is applied to give a simple criterion in terms of the parameters in the partitions p = 8f + 1 = X2+Y2 = C2+2D 2 for an odd prime q to be an octic residue (modp). Some consequences and a generalization of an analogous quartic residuacity law (proved by E. Lehmer in 1958) are also given.
... Since then, Scholz's reciprocity law has been proved using many different methods (see [3], [7], [10], and [14] for other proofs). The unfamiliar reader is referred to Emma Lehmer's expository article [9] for an overview of rational reciprocity laws and Williams, Hardy, and Friesen's article [15] for a proof of an all-encompassing rational quartic reciprocity law that was subsequently simplified by Evans [4] and Lemmermeyer [10]. ...
We provide a generalization of Scholz’s reciprocity law using the subfields K2t−1 and K2t of ℚ(ζp), of degrees 2t−1 and 2t over ℚ, respectively. The proof requires a particular choice of primitive element for K2t over K2t−1 and is based upon the splitting of the cyclotomic polynomial Φp(x) over the subfields.
... Lemmermeyer [1994] showed that when p ≡ 1 (mod 4), specific choices of A, B ∈ ޚ so that K 4 = (ޑ A + B √ p) result in the rational quartic reciprocity laws of Scholz [1934], Lehmer [1958;1978], and Burde [1969]. His work simplified the all-encompassing rational quartic reciprocity law of Williams et al. [1985] as well as its simplification by Evans [1989]. The reader unfamiliar with these laws may consult Lehmer's survey article [Lehmer 1978] and [Lemmermeyer 2000] for the relevant background. ...
The most natural extensions to the law of quadratic reciprocity are the rational reciprocity laws, described using the rational residue symbol. In this article, we provide a reciprocity law from which many of the known rational reciprocity laws may be recovered by picking appropriate primitive elements for subfields of Q. p/. As an example, a new generalization of Burde’s law is provided.
... In 1985, K. S. Williams, K. Hardy and C. Friesen [11] published a reciprocity formula that comprised all known rational quartic reciprocity laws. Their proof consisted in a long and complicated manipulation of Jacobi symbols and was subsequently simplified (and generalized) by R. Evans [3]. In this note we will give a proof of their reciprocity law which is not only considerably shorter but which also sheds some light on the raison d'être of rational quartic reciprocity laws. ...
We provide a simple proof of the general rational quartic reciprocity law due
to Williams, Hardy and Friesen.
We prove a general theorem that evaluates the Legendre symbol (A+Bm/p) under certain conditions on the integers A, B, m and the prime p. The evaluation is in terms of parameters appearing in a binary quadratic form representing p. The theorem has applications to quartic residuacity.
Let be the set of integers, and let p be a prime of the form . Suppose , , , , and . In this paper we establish congruences for and present new reciprocity laws.
Let ℤ be the set of integers, and let p be a prime of the form 4k + 1 and so p = c2 + d2 with c, d ∈ ℤ. Let q be an integer of the form 4k + 3. Assume that 4n2p = x2 + qy2 with c, d, n, x, y ∈ ℤ and (q, n) = (x, y) = 1, where (a, b) is the greatest common divisor of integers a and b. In this paper, we establish congruences for (-q)[p/8] (mod p) in terms of c, d, n, x and y, where [⋅] is the greatest integer function. In particular, we establish a reciprocity law and give an explicit criterion for (-11)[p/8] (mod p).
Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p≡1(mod4) be a prime, q∈ℤ, 2∤q and p=c 2 +d 2 =x 2 +qy 2 with c,d,x,y∈ℤ and c≡1(mod4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine q [p/8] (modp) in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.
Let be the set of integers, and let p be a prime of the form
4k+1. Suppose , , , , and . In this paper we continue to discuss congruences for
and present new reciprocity laws, but we assume
or , where is the greatest integer
function and .
Let be a prime, q be an odd number and for some integers c,d,x and y. Suppose that and c is coprime to x+d. In the paper, by using the quartic reciprocity law we determine in terms of c,d,x and y, where is the greatest integer function. When , we also determine . As an application we obtain the congruence for and the criterion for (if ), where is the Lucas sequence given by and . Hence we partially solve some conjectures posed by the author in previous two papers.
Let p ≡ 1 (mod 4) be a prime and a, b ∈ Z with a2 + b2 ≠ p. Suppose p = x2 + (a2 + b2) y2 for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine (frac(b + sqrt(a2 + b2), 2))frac(p - 1, 4) (mod p). As applications we obtain the congruences for Ufrac(p - 1, 4) modulo p and the criteria for p | Ufrac(p - 1, 8) (if p ≡ 1 (mod 8)), where {Un} is the Lucas sequence given by U0 = 0, U1 = 1 and Un + 1 = b Un + k2 Un - 1(n ≥ 1) . We also pose many conjectures concerning Ufrac(p - 1, 4), mfrac(p - 1, 8) or mfrac(p - 5, 8) (mod p).
Let be a prime. Let a,b∈Z with p∤a(a2+b2). In the paper we mainly determine by assuming p=c2+d2 or p=Ax2+2Bxy+Cy2 with AC−B2=a2+b2. As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=bUn+k2Un−1(n⩾1).
Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let εd be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that εd is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {un(a,b)} is the Lucas sequence defined by and .
This paper examines the e th power character of the divisors of two cyclotomic period polynomials of degree e 1 {e_1} and e 2 {e_2} . The special cases e 1 = 2 {e_1} = 2 and e 2 = 3 , 4 {e_2} = 3,4 , are considered in detail. As corollaries one finds new conditions for cubic and quartic residuacity. The computational method consists in representing cyclotomic numbers in the form c 1 ζ + c 2 ζ 2 + ⋯ + c p − 1 ζ p − 1 {c_1}\zeta + {c_2}{\zeta ^2} + \cdots + {c_{p - 1}}{\zeta ^{p - 1}} , where ζ = e 2 π i / p \zeta = {e^{2\pi i/p}} . Multiplication is reduced to addition and subtraction, which are carried out in a multi-precision system.
This paper examines the eth power character of the divisors of two cyclotomic period polynomials of degree and . The special cases and , are considered in detail. As corollaries one finds new conditions for cubic and quartic residuacity. The computational method consists in representing cyclotomic numbers in the form , where . Multiplication is reduced to addition and subtraction, which are carried out in a multi-precision system.
in the form L ≡ ±µM (mod q), and LM ≡ 0 (mod q).(Received March 07 1958)
The theory of cyclotomic period polynomials is developed for general periods of an arbitrary modulus, extending known results for the Gauss periods of prime modulus. Primes dividing the discriminant of the period polynomial are investigated, as are those primes dividing values of the period polynomial.
If k is a positive integer and p is a prime with p ≡ 1 (mod 2k), then 2(p-1)/2k is a 2kth root of unity modulo p. We consider the problem of determining 2(p-1)/2k modulo p. This has been done for k - 1, 2, 3 and the present paper treats k = 4 and 5, extending the work of Cunningham, Aigner, Hasse, and Evans.
A special prime q is a prime which divides the discriminant of a general period polynomial of degree e associated with the prime p = ef + 1, but q is neither an eth power residue oip nor a divisor of any value of this polynomial. These primes are very rare. Evans found some for the classical cyclotomic octic. There are none for lower degree cyclotomic polynomials. This paper finds special primes for the two quartics arising from the cyclotomy of Kloosterman sums for e = 8 and shows that there are none for e < 8.
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