Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.
We study real quadratic fields Q(D) such that, for a given rational integer m, all m-multiples of totally positive integers are sums of squares. We prove quite sharp necessary and sufficient ...conditions for this to happen. Further, we give a fast algorithm that verifies this property for specific m, D and we give complete results for m≤5000.
In this paper we mainly focus on some determinants with Legendre symbol entries. Let p be an odd prime and let (⋅p) be the Legendre symbol. We show that (−S(d,p)p)=1 for any d∈Z with (dp)=1, and ...that(Wpp)={(−1)|{0<k<p4:(kp)=−1}|ifp≡1(mod4),(−1)⌊(p+1)/8⌋ifp≡3(mod4), whereS(d,p)=det(i2+dj2p)1⩽i,j⩽(p−1)/2 andWp=det(i2−((p−1)/2)!jp)0⩽i,j⩽(p−1)/2. We also pose some conjectures on determinants, one of which states that (−1)⌊(p+1)/8⌋Wp is a square when p≡3(mod4).
Let k be a real quadratic number field, and k∞ its cyclotomic Z2-extension. We study the cyclicity of the Galois group X∞′ over k∞ of the maximal abelian unramified 2-extension, in which all 2-adic ...primes of k∞ split completely. As consequence, we determine the complete list of real quadratic number fields for which X∞′ is cyclic.
When X∞′ is cyclic non-trivial, we give a new infinite family of real quadratic number fields, for which Greenberg's conjecture is valid.
In this paper we establish a connection between the Gauss factorials and Iwasawa's cyclotomic λ-invariant for an imaginary quadratic field K. As a result, we will explain a correspondence between the ...1-exceptional primes of Cosgrave and Dilcher 2, 3 for m=3 and m=4, and the primes for which the λ-invariants for K=Q(−3) and K=Q(i) are greater than one, respectively. We refer to the latter primes as “non-trivial” for their respective fields. We will also see that similar correspondences are true for K=Q(−d) when d=2,5 and 6. As a corollary we find that primes p of the form p2=3x2+3x+1 are always non-trivial for K=Q(−3). Last, we show that the non-trivial primes p for K=Q(i) and K=Q(−3) are characterized by modulo p2 congruences involving Euler and Glaisher numbers respectively.
Abstract Let $n$ be an integer congruent to $0$ or $3$ modulo $4$ . Under the assumption of the ABC conjecture, we prove that, given any integer $m$ fulfilling only a certain coprimeness condition, ...there exist infinitely many imaginary quadratic fields having an everywhere unramified Galois extension of group $A_n \times C_m$ . The same result is obtained unconditionally in special cases.
In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of ...how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.
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