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Let p ≥ 5 be a prime number such that p ≡ 3 (mod 8) and d an odd positive squarefree integer. Put K = Q (2 , - p , d) . In this paper, we compute the 2-rank of the class group of the fields K = Q (2 , - p , d) . Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic and of type (2, 2) or (2, 4). [ABSTRACT FROM AUTHOR]

We state and prove a Gras-type equality relating a certain module generated by Rubin–Stark units and the ray class number. We use for this purpose the notion of generalized index. [ABSTRACT FROM AUTHOR]

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one.

Let be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and its second Hilbert 2-class field. Denote by the Galois group of . The purpose of this note is to investigate the Hilbert 2-class field tower of and then deduce the structure of G. [ABSTRACT FROM AUTHOR]

Let ε be an algebraic unit of the degree n ⩾ 3. Assume that the extension ℚ(ε)/ℚ is Galois. We would like to determine when the order ℤ[ε] of ℚ(ε) is Gal(ℚ(ε)/ℚ)-invariant, i.e. when the n complex conjugates ε1, ..., εn of ε are in ℤ[ε], which amounts to asking that ℤ[ε1, ..., εn] = ℤ[ε], i.e., that these two orders of ℚ(ε) have the same discriminant. This problem has been solved only for n = 3 by using an explicit formula for the discriminant of the order ℤ[ε1, ε2, ε3]. However, there is no known similar formula for n > 3. In the present paper, we put forward and motivate three conjectures for the solution to this determination for n = 4 (two possible Galois groups) and n = 5 (one possible Galois group). In particular, we conjecture that there are only finitely many cyclic quartic and quintic Galois-invariant orders generated by an algebraic unit. As a consequence of our work, we found a parametrized family of monic quartic polynomials in ℤ[X] whose roots ε generate bicyclic biquadratic extensions ℚ(ε)/ℚ for which the order ℤ[ε] is Gal(ℚ(ε)/ℚ)-invariant and for which a system of fundamental units of ℤ[ε] is known. According to the present work it should be difficult to find other similar families than this one and the family of the simplest cubic fields. [ABSTRACT FROM AUTHOR]

In this article, we show that certain abelian extensions K with unit rank greater than or equal to three have cyclic class group if and only if it has a Euclidean ideal class. This result improves an earlier result of Murty and Graves. One can improve this result up to unit rank 2 if one assumes the Elliott and Halberstam conjecture (see Conjecture 1 in preliminaries). These results are known under generalized Riemann hypothesis by the work of Lenstra (J Lond Math Soc 10:457–465) [see also Weinberger (Proc Symp Pure Math 24:321–332)]. [ABSTRACT FROM AUTHOR]

Let K be a number field. In the terminology of Nagell a unit ε of K is called exceptional if 1 − ε is also a unit. The existence of such a unit is equivalent to the fact that the unit equation ε 1 + ε 2 + ε 3 = 0 is solvable in units ε 1 , ε 2 , ε 3 of K . Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer k with k ≥ 3 , denoted by ℓ ( K ) , such that the unit equation ε 1 + … + ε k = 0 is solvable in units ε 1 , … , ε k of K . If no such k exists, we set ℓ ( K ) = ∞ . Apart from trivial cases when ℓ ( K ) = ∞ , we give an explicit upper bound for ℓ ( K ) . We obtain several results for ℓ ( K ) in number fields of degree at most 4, cyclotomic fields and general number fields of given degree. We prove various properties of ℓ ( K ) , including its magnitude, parity as well as the cardinality of number fields K with given degree and given odd resp. even value ℓ ( K ) . Finally, as an application, we deal with certain arithmetic graphs, namely we consider the representability of cycles. We conclude the paper by listing some problems and open questions. [ABSTRACT FROM AUTHOR]

For a knot K in a homology 3-sphere Σ, let M be the result of 2 / q -surgery on K , and let X be the universal abelian covering of M . Our first theorem is that if the first homology of X is finite cyclic and M is a Seifert fibered space with N ≥ 3 singular fibers, then N ≥ 4 if and only if the first homology of the universal abelian covering of X is infinite. Our second theorem is that under an appropriate assumption on the Alexander polynomial of K , if M is a Seifert fibered space, then q = ± 1 (i.e. integral surgery). [ABSTRACT FROM AUTHOR]

Let O K and C K be respectively the ring of integers and the class group of a number field K . For each integer q ≥ 2 , denote by ∏ q ( K ) the product of all the maximal ideals of O K with norm q , if these ideals do not exist we set ∏ q ( K ) = O K . The Pólya group of K is the subgroup of C K generated by the classes of the ideals ∏ q ( K ) , and K is called a Pólya field if the module of integer-valued polynomials over O K has a regular basis. In this paper, we determine Pólya group of any imaginary bicyclic biquadratic number field, and thus we deduce all the imaginary bicyclic biquadratic Pólya fields. [ABSTRACT FROM AUTHOR]

Let G be an additive finite abelian group. For a positive integer k , let s ≤ k ( G ) denote the smallest integer l such that each sequence of length l has a non-empty zero-sum subsequence of length at most k . Among other results, we determine s ≤ k ( G ) for all finite abelian groups of rank two. [ABSTRACT FROM AUTHOR]

We compute the p -adic regulator of cyclic cubic extensions of Q with discriminant up to 10 16 for 3 < p < 100 , and observe the distribution of the p -adic valuation of the regulators. We find that for almost all primes, the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions. Based on this random matrix model, a conjecture on the distribution of the valuations of p -adic regulators of cyclic cubic fields is stated. [ABSTRACT FROM AUTHOR]

The set of distances of a monoid or of a domain is the set of all d ∈ N with the following property: there are irreducible elements u 1 , … , u k , v 1 , … , v k + d such that u 1 ⋅ … ⋅ u k = v 1 ⋅ … ⋅ v k + d , but u 1 ⋅ … ⋅ u k cannot be written as a product of l irreducible elements for any l with k < l < k + d . We show that the set of distances is an interval for certain seminormal weakly Krull monoids which include seminormal orders in holomorphy rings of global fields. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with char F q = p and n > 0 an integer with gcd ( n , log p ⁡ q ) = 1 . Let ( ) ⁎ : F q ( x 0 , … , x n − 1 ) → F q ( x 0 , … , x n − 1 ) be the F q -monomorphism defined by x i ⁎ = x i + 1 for 0 ≤ i < n − 1 and x n − 1 ⁎ = x 0 q . For f , g ∈ F q ( x 0 , … , x n − 1 ) ∖ F q , define f ∘ g = f ( g , g ⁎ , … , g ( n − 1 ) ⁎ ) . Then ( F q ( x 0 , … , x n − 1 ) ∖ F q , ∘ ) is a monoid whose invertible elements are called global P -forms. Global P -forms were first introduced by H. Dobbertin in 2001 with q = 2 to study a certain type of permutation polynomials of F 2 m with gcd ( m , n ) = 1 ; global P -forms with q = p for an arbitrary prime p were considered by W. More in 2005. In this paper, we discuss some fundamental questions about global P -forms, some of which are answered and others remain open. [ABSTRACT FROM AUTHOR]

Inspired by the Odlyzko root discriminant and Golod–Shafarevich p -group bounds, Martinet (1978) asked whether an imaginary quadratic number field K / Q must always have an infinite Hilbert 2-class field tower when the class group of K has 2-rank 4, or equivalently when the discriminant of K has 5 prime factors. No negative results are known. Benjamin (2001, 2002) and Sueyoshi (2004, 2009, 2010) systematically established infinite 2-towers for many K in question, by casework on the associated Rédei matrices. Others, notably Mouhib (2010), have also made progress, but still many cases remain open, especially when the class group of K has small 4-rank. Recently, Benjamin (2015) made partial progress on several of these open matrices when the class group of K has 4-rank 1 or 2. In this paper, we partially address many open cases when the 4-rank is 0 or 2, affirmatively answering some questions of Benjamin. We then investigate barriers to our methods and ask an extension question (of independent interest) in this direction. Finally, we suggest places where speculative refinements of Golod–Shafarevich or group classification methods might overcome the “near miss” inadequacies in current methods. [ABSTRACT FROM AUTHOR]

For a knot K with Δ K ( t ) ≐ t 2 − 3 t + 1 in a homology 3-sphere, let M be the result of 2 / q -surgery on K . We show that appropriate assumptions on the Reidemeister torsion and the Casson–Walker–Lescop invariant of the universal abelian covering of M imply q = ± 1 , if M is a Seifert fibered space. [ABSTRACT FROM AUTHOR]

Let ϵ be a totally real cubic algebraic unit. Assume that the cubic number field Q ( ϵ ) is Galois. In this situation, it is natural to ask when the cubic order Z [ ϵ ] is invariant under the action of the Galois group Gal ( Q ( ϵ ) / Q ) . It seems that this natural problem has never been looked at. We give an answer to this problem (e.g., we show that if ϵ is totally positive, then this happens in only 12 cases). [ABSTRACT FROM AUTHOR]

We give a signed fundamental domain for the action on C ⁎ × R + n − 2 of the totally positive units E ( k ) + of a number field k of degree n and having exactly one pair of complex embeddings. This signed fundamental domain, built of k -rational simplicial cones, is as convenient as a true fundamental domain for the purpose of studying Dedekind zeta functions. However, while there is no general construction of a true fundamental domain, we construct a signed fundamental domain from any set of fundamental units of k . [ABSTRACT FROM AUTHOR]

Let F denote a number field. We study a relation between the subgroup of elements whose l th roots generate extensions of F which are contained in a Z l -extension of F and a certain kernel of Milnor's K -group defined by Tate. We prove that both groups can be described in terms of a norm compatible sequence over the cyclotomic Z l -extension of F . [ABSTRACT FROM AUTHOR]

We formulate a conjecture which generalizes Darmon’s ‘refined class number formula’. We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the ‘except $2$-part’ of Darmon’s conjecture, which was first proved by Mazur and Rubin. [ABSTRACT FROM AUTHOR]

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We study lens space surgeries along two different families of 2-component links, denoted by ${A}_{m, n} $ and ${B}_{p, q} $, related with the rational homology $4$-ball used in J. Park’s (generalized) rational blow down. We determine which coefficient $r$ of the knotted component of the link yields a lens space by Dehn surgery. The link ${A}_{m, n} $ yields a lens space only by the known surgery with $r= mn$ and unexpectedly with $r= 7$ for $(m, n)= (2, 3)$. On the other hand, ${B}_{p, q} $ yields a lens space by infinitely many $r$. Our main tool for the proof are the Reidemeister-Turaev torsions, that is, Reidemeister torsions with combinatorial Euler structures. Our results can be extended to the links whose Alexander polynomials are same as those of ${A}_{m, n} $ and ${B}_{p, q} $. [ABSTRACT FROM PUBLISHER]