Biquadratic Fields Research Articles

Large Pólya groups in simplest cubic fields and consecutive bi-quadratic fields

The Pólya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large class numbers by Cherubini et al., we extend the notion of consecutiveness of number fields to certain parametric families of cyclic cubic fields and bi-quadratic fields and address the question of the existence of infinitely many such consecutive fields with large Pólya groups. This extends a recent result of the second author and Saikia for totally real bi-quadratic fields.

Minimal rank of universal lattices and number of indecomposable elements in real multiquadratic fields

We establish an upper bound on the number of real multiquadratic fields that admit a universal quadratic lattice of a given rank, or contain a given amount of indecomposable elements modulo totally positive units, obtaining density zero statements. We also study the structure of indecomposable elements in real biquadratic fields, and compute a system of indecomposable elements modulo totally positive units for some families of real biquadratic fields.

Sums of squares of integral multiples of an integral element of real bi-quadratic fields

Sums of squares of integral multiples of an integral element of real bi-quadratic fields

Correction to: On Greenberg’s conjecture for certain real biquadratic fields

Correction to: On Greenberg’s conjecture for certain real biquadratic fields

Biquadratic magnetic coupling effect in CoPt/Cr/Fe90Co10 orthogonal structures

In this work, we present the biquadratic field H bq contribution to increase a frequency of spin-torque oscillation (STO) in an orthogonal magnetization structure in simulation, and realize such an orthogonal structure by preparing Co/Pt lamination as the bottom perpendicular magnetic anisotropy layer, Cr or Cu as the spacer, and experimentally realize Fe90Co10 as the top free layer. Our observations of the Cr-spacer sample reveal a notable challenge in achieving magnetic saturation, underscoring the role of H bq in suppressing magnetization reversal and its potential to broaden the STO current range and increase the STO frequency. This leads to the manifestation of spin-transfer-torque oscillations in an orthogonal structure, bolstered by robust biquadratic magnetic coupling, thus attaining high and stable STOs in the simulations.

Lifting problem for universal quadratic forms over totally real cubic number fields

AbstractLifting problem for universal quadratic forms asks for totally real number fields that admit a positive definite quadratic form with coefficients in that is universal over the ring of integers of . In this paper, we show is the only such totally real cubic field. Moreover, we show that there is no such biquadratic field.

Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields

Quaternions are extensions of complex numbers that are four-dimensional objects. Quaternion consists of one real number and three complex numbers, commonly denoted by the standard vectors and . Quaternion algebra over the field is an algebra in which the multiplication between standard vectors is non-commutative and the multiplication of standard vector with itself is a member of the field. The field considered in this study is the quadratic field and its extensions are biquadratic and composite. There have been many studies done to show the existence of split properties in quaternion algebras over quadratic fields. The purpose of this research is to prove a theorem about the existence of split properties on three field structures, namely quaternion algebras over quadratic fields, biquadratic fields, and composite of quadratic fields. We propose two theorems about biquadratic fields and composite of quadratic fields refer to theorems about the properties of the split on quadratic fields. The result of this research is a theorem proof of three theorems with different field structures that shows the different conditions of the three field structures. The conclusion is that the split property on quaternion algebras over fields exists if certain conditions can be met.

Non-Pólya bi-quadratic fields with an Euclidean ideal class

In this paper, we prove the existence of three pairwise distinct families of totally real bi-quadratic fields, each having Pólya group isomorphic to Z/2Z. This extends the previously known families of number fields considered by Heidaryan and Rajaei. Our results also establish that under mild assumptions, the possibly infinite families of bi-quadratic fields having a non-principal Euclidean ideal class, considered by Chattopadhyay and Muthukrishnan, fail to be Pólya fields.

Class numbers of certain families of totally real biquadratic fields and a result of Mollin

Class numbers of certain families of totally real biquadratic fields and a result of Mollin

Non-principal Euclidean ideal class in a family of biquadratic fields with the class number two

Non-principal Euclidean ideal class in a family of biquadratic fields with the class number two

The geometry of the moduli space of non-cyclic biquadratic field extensions

In this paper, we investigate the existence of an elementary abelian closure in characteristic not 2 for biquadratic extensions. We discover that it exists for any non-cyclic extension. We make use of it to obtain a classification for this class of extensions up to isomorphism via descent. This permits us to describe the geometry of this moduli space in group theoretic terms. We also provide two families of polynomials with two parameters that can describe any quartic extensions.

Pythagoras numbers of orders in biquadratic fields

We examine the Pythagoras number P(OK) of the ring of integers OK in a totally real biquadratic number field K. We show that the known upper bound 7 is attained in a large and natural infinite family of such fields. In contrast, for almost all fields Q(5,s) we prove P(OK)=5. Further we show that 5 is a lower bound for all but seven fields K and 6 is a lower bound in an asymptotic sense.

Totally real bi-quadratic fields with large Pólya groups

For an algebraic number field K with ring of integers $$\mathcal {O}_{K}$$ , an important subgroup of the ideal class group $$Cl_{K}$$ is the Pólya group, denoted by Po(K), which measures the failure of the $$\mathcal {O}_{K}$$ -module $$Int(\mathcal {O}_{K})$$ of integer-valued polynomials on $$\mathcal {O}_{K}$$ from admitting a regular basis. In this paper, we prove that for any integer $$n \ge 2$$ , there are infinitely many totally real bi-quadratic fields K with $$Po(K) \simeq ({\mathbb {Z}}/2{\mathbb {Z}})^{n}$$ . In fact, we explicitly construct such an infinite family of number fields. This also provides an infinite family of bi-quadratic fields with ideal class groups of 2-ranks at least n.

Euclidean algorithm in Galois quartic fields

We prove that all imaginary biquadratic fields and cyclic quartic fields of class number 1 are Euclidean.

Kuroda’s formula and arithmetic statistics

Kuroda's formula relates the class number of a multi-quadratic number field $K$ to the class numbers of its quadratic subfields $k_i$. A key component in this formula is the unit group index $Q(K) = [\mathcal{O}_{K}^{\times}: \prod_i\mathcal{O}_{k_i}^{\times}]$. We study how $Q(K)$ behaves on average in certain natural families of totally real biquadratic fields $K$ parametrized by prime numbers.

Pólya group in some real biquadratic fields

For a Galois extension K/Q, Pólya group Po(K) of K coincides with the subgroup of strongly ambiguous ideal classes of K, which is controlled by ramification and Galois cohomology. Using Setzer's result [9] on the first cohomology group of units in the real biquadratic fields, we describe the structure of Po(K) for an infinite family of real biquadratic fields K with five ramifications and no quadratic Pólya subfield. Moreover, #Po(K) in this family is the same as the Hasse unit index.

2-Class groups of cyclotomic towers of imaginary biquadratic fields and applications

Let [Formula: see text] be an odd positive square-free integer. In this paper, we shall investigate the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of the imaginary biquadratic number field [Formula: see text] if [Formula: see text] is of specifiic form. Furthermore, we deduce the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of [Formula: see text].

First-Degree Prime Ideals of Biquadratic Fields Dividing Prescribed Principal Ideals

We describe first-degree prime ideals of biquadratic extensions in terms of the first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. The correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of specially-shaped principal ideals in their respective rings, with some exceptions that we explicitly characterize.

Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s $p$-rationality conjecture

In this paper we make a series of numerical experiments to support Greenberg's p-rationality conjecture, we present a family of p-rational biquadratic fields and we find new examples of p-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the p-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.

There are no universal ternary quadratic forms over biquadratic fields

AbstractWe study totally positive definite quadratic forms over the ring of integers$\mathcal {O}_K$of a totally real biquadratic field$K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in$2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of$\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of$\mathcal {O}_K$; we prove several new results about their properties.