Quartic Extensions Research Articles

Quadratic polynomials represented by norm forms

The Hasse principle and weak approximation is established for equations of the shape P(t)=N(x_1,x_2,x_3,x_4), where P is an irreducible quadratic polynomial in one variable and N is a norm form associated to a quartic extension of the rationals containing the roots of P. The proof uses analytic methods.

HT90 and “simplest” number fields

A standard formula (1.1) leads to a proof of HT90, but requires proving the existence of $\theta$ such that $\alpha\ne0$, so that $\beta=\alpha/\sigma(\alpha)$. We instead impose the condition (M), that taking $\theta=1$ makes $\alpha=0$. Taking $n=3$, we recover Shanks’s simplest cubic fields. The “simplest” number fields of degrees $3$ to $6$, Washington’s cyclic quartic fields, and a certain family of totally real cyclic extensions of $\mathbb{Q} (\cos(\pi/4m))$ all have defining polynomials whose zeroes satisfy (M). Further investigation of (M) for $n=4$ leads to an elementary algebraic construction of a $2$-parameter family of octic polynomials with “generic” Galois group ${}_{8}T_{11}$. Imposing an additional algebraic condition on these octics produces a new family of cyclic quartic extensions. This family includes the “simplest” quartic fields and Washington’s cyclic quartic fields as special cases. We obtain more detailed results on our octics when the parameters are algebraic integers in a number field. In particular, we identify certain sets of special units, including exceptional sequences of $3$ units, and give some of their properties.

Parafermions for higher order extensions of the Poincaré algebra and their associated superspace

Parafermions of orders 2 and 3 are shown to be the fundamental tool to construct superspaces related to cubic and quartic extensions of the Poincaré algebra. The corresponding superfields are constructed, and some of their main properties are analyzed in detail. In this context, the existence problem of operators acting like covariant derivatives is analyzed, and the associated operators are explicitly constructed.

A Weil Descent Attack against Elliptic Curve Cryptosystems over Quartic Extension Fields

This paper proposes a Weil descent attack against elliptic curve cryptosystems over quartic extension fields. The scenario of the attack is as follows: First, one reduces a DLP on a Weierstrass form over the quartic extention of a finite field k to a DLP on a special form, called Scholten form, over the same field. Second, one reduces the DLP on the Scholten form to a DLP on a genus two hyperelliptic curve over the quadratic extension of k. Then, one reduces the DLP on the hyperelliptic curve to one on a Cab model over k. Finally, one obtains the discrete-log of original DLP by applying the Gaudry method to the DLP on the Cab model. In order to carry out the scenario, this paper shows that many of elliptic curve discrete-log problems over quartic extension fields of odd characteristics are reduced to genus two hyperelliptic curve discrete-log problems over quadratic extension fields, and that almost all of the genus two hyperelliptic curve discrete-log problems over quadratic extension fields of odd characteristics come under Weil descent attack. This means that many of elliptic curve cryptosystems over quartic extension fields of odd characteristics can be attacked uniformly.

Counting cyclic quartic extensions of a number field

In this paper, we give asymptotic formulas for the number of cyclic quartic extensions of a number field.

The Witt kernels of purely inseparable quartic extensions

For a purely inseparable quartic field extension L/ k, we determine the Witt kernel W ( L/ k) of quadratic k-forms that split hyperbolically over L. In particular, we show that W ( L/ k) is generated (as a W ( k)-module) by quadratic Pfister forms of dimension four.

A Nonlinear Elastic Beam System with Inelastic Contact Constraints

Abstract. In this paper we study freely propagating inertial, i.e., unforced, waves, in an elastic beam constrained so that all motion takes place above and on a flat, rigid support surface, subject to a gravitational force and a compressive longitudinal load. Contact between the beam and the support surface is assumed to be completely inelastic. A nonlinear beam model is used, incorporating a quartic extension of the familiar quadratic potential energy functional for the standard Euler—Bernoulli model. After briefly reviewing the rationale for the model and some of its properties, as developed in earlier articles, we present existence and uniqueness results for the constrained system obtained with the use of a ``penalty function'' approach involving the addition of a ``uni-directional friction'' dissipative term, active only when the constraint is violated, to the unconstrained system.

Enumerating Quartic Dihedral Extensions of ℚ

We give an explicit Dirichlet series for the generating function of the discriminants of quartic dihedral extensions of ℚ. From this series we deduce an asymptotic formula for the number of isomorphism classes of such quartic extensions with discriminant up to a given bound. On the other hand, by using essentially classical results of genus theory combined with elementary analytical methods such as the method of the hyperbola, we show how to compute exactly this number up to quite large bounds, and we give a table of selected values.

Fast exponentiation in subgroups of finite fields

A new method for fast exponentiation is introduced by adapting the work of Gallant, Lambert and Vanstone (see Advances in Cryptology-Crypto 2001 LNCS 2139, p.190-200 Springer-Verlag, 2001) to finite fields: thus, it is applicable to the multiplicative subgroups of non-prime fields with efficient group homomorphisms such as pth powering maps. When applied to a quartic extension of a 256 bit prime field, the proposed method runs up to 60% faster than ordinary exponentiation in a 1024 bit prime field.

Computing all monogeneous mixed dihedral quartic extensions of a quadratic field

Let M be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields K with mixed signature having power integral bases and containing M as a subfield. We also determine all generators of power integral bases in K. Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for M=ℚ(2),ℚ(3),ℚ(5).

Computing Power Integral Bases in Quartic Relative Extensions

We develop an algorithm for computing all generators of relative power integral bases in quartic extensions K of number fields M. For this purpose we use the main ideas of our previously derived algorithm for solving index form equations in quartic fields (I. Gaál, A. Pethő, and M. Pohst, 1993, J. Symbolic Comput.16, 563–584; 1996, J. Number Theory57, 90–104). In this way we reduce the problem to the resolution of a cubic and several corresponding quartic relative Thue equations over M. These equations determine the generators of power integral bases of K over M up to translation by integers of M and multiplication by unit factors of M. The new method is based on our ability to solve relative Thue equations efficiently by the algorithm in (I. Gaál and M. Pohst, 2000, Math. Comp., to appear). In the case K is an octic field with a quadratic subfield M we can also consider the absolute index of elements of K, having relative index 1 over M. In order to determine all generators of power integral bases of K (over Q) we determine the corresponding translating elements and unit factors properly. This is done by solving an equation similar to an inhomogeneous Thue equation. We illustrate our algorithms with detailed examples.

Quartic Extensions of &#8474

Quartic Extensions of &#8474

The conductor of a cyclic quartic field using Gauss sums

Let Q denote the field of rational numbers. Let K be a cyclic quartic extension of Q. It is known that there are unique integers A, B, C, D such that \(K = Q\left( {\sqrt {A(D + B\sqrt D )} } \right)\) where A is squarefree and odd, D=B2+C2 is squarefree, B\( > \) 0 , C\( > \) 0, GCD(A,D)=1. The conductor f(K) of K is f(K) = 2l|A|D, where \(l = \left\{ \begin{gathered} 3,{\text{ if }}D \equiv 2{\text{ }}({\text{mod 4}}){\text{ or }}D \equiv 1{\text{ (mod 4), }}B \equiv 1{\text{ }}({\text{mod 2}}), \hfill 2,{\text{ if }}D \equiv 1{\text{ (mod 4), }}B \equiv 0{\text{ (mod 2), }}A + B \equiv 3{\text{ (mod 4),}} \hfill 0,{\text{ if }}D \equiv 1{\text{ (mod 4), }}B \equiv 0{\text{ (mod 2), }}A + B \equiv 1{\text{ (mod 4)}}{\text{.}} \hfill \end{gathered} \right.\) A simple proof of this formula for f(K) is given, which uses the basic properties of quartic Gauss sums.

Determination of all Quaternion Octic CM-Fields with Class Number 2

It is known that there are only finitely many normal CM-fields with class number one or with given class number (see [9, Theorem 2; 11, Theorem 2]) and J. Hoffstein showed that the degree of any normal CMfield with class number one is less than 436 (see [2, Corollary 2]). Moreover, K. Yamamura has determined all the abelian CM-fields with class number one: there are 172 non-isomorphic such number fields. In a recent paper the author and R. Okazaki moved on to the determination of non-abelian but normal octic CM-fields with class number one. Noticing that their class numbers are always even, they got rid of quaternion octic CM-fields, then they focussed on dihedral octic CM-fields and proved that there are 17 dihedral octic CM-fields with class number one. The aim of this paper is to get back to the quaternion case: we shall show that there exists exactly one quaternion octic CM-field with class number 2, namely: Q(V<x) wi tha = ( 2 + y'2)(3 + V3). Moreover, we shall show that the Hilbert class field of this number field is a normal and non-abelian CM-field of degree 16 with class number one. 1. Factorization of the Dedekind zeta function of a quaternion octic number field Let N be a quaternion octic CM-field, that is, N/Q is a normal octic extension with Galois group Gal (N/Q) the quaternion group Q8 with eight elements Q8 = {± 1, ± U ±j, ± k} with i = / = k = — 1 and ij = —ji = kjk = — kj = i and hi = — ik = j . Let N be the quartic subfield of N, so that N is the maximal totally real subfield of N. Let ki5 k and kfc be the three quadratic subfields of N, so that these quadratic fields are real quadratic fields. The quartic extensions N/k4, N/k; and N/kfc are cyclic and the quartic extension N/Q is biquadratic bicyclic. The lattice of subfields can be set out as follows:

Families of Non-Galois Quartic Fields

In this article we compute a basis of the ring of integers and the group of units of some quartic extensions of Q given by the first coordinate of a torsion point of an elliptic curve over Q.

Biquaternion algebras and quartic extensions

Biquaternion algebras and quartic extensions

Steinitz classes of order 2 in quadratic and quartic fields

For a given number field K, does there exist an extension M of odd prime degree l such that the relative discriminant of M/K is a principal ideal, but M/K has no relative integral basis? A general, but incomplete answer is given to this question when K/Q is a normal extension. If, in addition, [ K: Q] is odd, the answer is complete. A detailed study is done when K/Q is a quadratic or normal quartic extension.

𝐿-functions and class numbers of imaginary quadratic fields and of quadratic extensions of an imaginary quadratic field

Starting from the analytic class number formula involving its L-function, we first give an expression for the class number of an imaginary quadratic field which, in the case of large discriminants, provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms, as has been used by Buell in order to compute his class number tables. Then, using class field theory, we will construct a periodic character χ \chi , defined on the ring of integers of a field K that is a quadratic extension of a principal imaginary quadratic field k, such that the zeta function of K is the product of the zeta function of k and of the L-function L ( s , χ ) L(s,\chi ) . We will then determine an integral representation of this L-function that enables us to calculate the class number of K numerically, as soon as its regulator is known. It will also provide us with an upper bound for these class numbers, showing that Hua’s bound for the class numbers of imaginary and real quadratic fields is not the best that one could expect. We give statistical results concerning the class numbers of the first 50000 quadratic extensions of Q ( i ) {\mathbf {Q}}(i) with prime relative discriminant (and with K/Q a non-Galois quartic extension). Our analytic calculation improves the algebraic calculation used by Lakein in the same way as the analytic calculation of the class numbers of real quadratic fields made by Williams and Broere improved the algebraic calculation consisting in counting the number of cycles of reduced ideals. Finally, we give upper bounds for class numbers of K that is a quadratic extension of an imaginary quadratic field k which is no longer assumed to be of class number one.

Rigidity of Decomposition laws and number fields

AbstractWe speak of rigidity, if partial information about the prime decomposition in an extension of number fields K¦k determines the decomposition law completely (and hence the zeta function ζK), or even fixes the field K itself. Several concepts of rigidity, depending on the degree of information we start from, are introduced and studied. The strongest concept (absolute rigidity) was only known to hold for the ground field and all quadratic extensions. Here a complete list of all Galois quartic extensions which are absolutely rigid is given. For the weaker concept of rigidity, all rigid situations among the fields of degree up to 8 are determined.

Quadratic subfields on quartic extensions of local fields

We show that any quartic extension of a local field of odd residue characteristic must contain an intermediate field. A consequence of this is that local fields of odd residue characteristic do not have extensions with Galois group A4 or S4. Counterexamples are given for even residue characteristic.