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Different groups of circular units of a compositum of real quadratic fields

Radan Kučera (1994)

Acta Arithmetica

There are many different definitions of the group of circular units of a real abelian field. The aim of this paper is to study their relations in the special case of a compositum k of real quadratic fields such that -1 is not a square in the genus field K of k in the narrow sense. The reason why fields of this type are considered is as follows. In such a field it is possible to define a group C of units (slightly bigger than Sinnott's group of circular units) such that the Galois...

Dihedral and cyclic extensions with large class numbers

Peter J. Cho, Henry H. Kim (2012)

Journal de Théorie des Nombres de Bordeaux

This paper is a continuation of [2]. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups, dihedral groups D n , n = 3 , 4 , 5 , and cyclic groups C n , n = 4 , 5 , 6 . We first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modification of zero density result of Kowalski-Michel, we choose subfamilies such that the corresponding L -functions are zero free...

Diophantine equations and class number of imaginary quadratic fields

Zhenfu Cao, Xiaolei Dong (2000)

Discussiones Mathematicae - General Algebra and Applications

Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and μ i ∈ - 1 , 1 ( i = 1 , 2 ) , and let h ( - 2 1 - e D ) ( e = 0 o r 1 ) denote the class number of the imaginary quadratic field ℚ ( √ ( - 2 1 - e D ) ) . In this paper, we give the all-positive integer solutions of the Diophantine equation Ax² + μ₁B = K((Ay² + μ₂B)/K)ⁿ, 2 ∤ n, n > 1 and we prove that if D > 1, then h ( - 2 1 - e D ) ≡ 0 ( m o d n ) , where D, and n satisfy k ⁿ - 2 e + 1 = D x ² , x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.

Discriminants of Chebyshev radical extensions

T. Alden Gassert (2014)

Journal de Théorie des Nombres de Bordeaux

Let t be any integer and fix an odd prime ℓ . Let Φ ( x ) = T ℓ n ( x ) - t denote the n -fold composition of the Chebyshev polynomial of degree ℓ shifted by t . If this polynomial is irreducible, let K = ℚ ( θ ) , where θ is a root of Φ . We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on t that ensure K is monogenic. For other values of t , we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of K and compute an integral basis for the ring of integers...

Currently displaying 101 – 120 of 146