Biquadratic Fields Research Articles

Hilbert genus fields of real biquadratic fields

The Hilbert genus field of the real biquadratic field $$K=\mathbb {Q}(\sqrt{\delta },\sqrt{d})$$ is described by Yue (Ramanujan J 21:17–25, 2010) and by Bae and Yue (Ramanujan J 24:161–181, 2011) explicitly in the case $$\delta =2$$ or $$p$$ with $$p\equiv 1 \, \mathrm{mod}\, 4$$ a prime and $$d$$ a squarefree positive integer. In this article, we describe explicitly the case that $$\delta =p, 2p$$ or $$p_1p_2$$ where $$p$$ , $$p_1$$ , and $$p_2$$ are primes congruent to $$3$$ modulo $$4$$ , and $$d$$ is any squarefree positive integer, thus complete the construction of the Hilbert genus field of real biquadratic field $$K=K_0(\sqrt{d})$$ such that $$K_0=\mathbb {Q}(\sqrt{\delta })$$ has an odd class number.

Structure of $$\mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k})$$ Gal ( k 2 ( 2 ) / k ) for some fields $$\mathbb {k}=\mathbb {Q}(\sqrt{2p_1p_2},i)$$ k = Q ( 2 p 1 p 2 , i ) with $$\mathbf {C}l_2(\mathbb {k})\simeq (2, 2, 2)$$ C l 2 ( k ) ≃ ( 2 , 2 , 2 )

Let $p_1 \equiv p_2 \equiv5\pmod8$ be different primes. Put $i=\sqrt{-1}$ and $d=2p_1p_2$, then the bicyclic biquadratic field $k=Q(\sqrt{d}, \sqrt{-1})$ has an elementary abelian 2-class group of rank $3$. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group $\mathrm{Gal}(k_2^{(2)}/k)$ of the second Hilbert 2-class field $k_2^{(2)}$ of $k$. We study the capitulation problem of the 2-classes of $k$ in its seven unramified quadratic extensions $K_i$ and in its seven unramified bicyclic biquadratic extensions $L_i$.

ON THE STRONGLY AMBIGUOUS CLASSES OF 𝕜/ℚ(i) WHERE $\mathds{k} = {\mathbb Q}(\sqrt{2p_{1}p_{2}, i})$

We construct an infinite family of imaginary bicyclic biquadratic number fields 𝕜 with the 2-ranks of their 2-class groups are ≥ 3, whose strongly ambiguous classes of 𝕜/ℚ(i) capitulate in the absolute genus field 𝕜(*), which is strictly included in the relative genus field (𝕜/ℚ(i))* and we study the capitulation of the 2-ideal classes of 𝕜 in its quadratic extensions included in 𝕜(*).

Sums of three integral squares in biquadratic fields

TextLet K be an algebraic number field and OK the ring of integers in K. Let SK be the set of all elements α∈OK which are sums of squares in OK and s(OK) the minimal number of squares necessary to represent −1 in OK. Let g(SK) be the smallest positive integer t such that every element in SK is a sum of t squares in OK. In this note, for K=Q(−m,−n), where m≡n≡3(mod4) are two distinct positive square-free integers, we show that SK=OK and if s(OK)=2, then g(OK)=3. VideoFor a video summary of this paper, please click here or visit http://youtu.be/F6UHHgzfdzA.

CLASS NUMBER FORMULAS FOR CERTAIN BICYCLIC BIQUADRATIC NUMBER FIELDS AS FINITE SUMS INVOLVING JACOBI ELLIPTIC FUNCTIONS

We give formulas for the class numbers of bicyclic biquadratic number fields containing an imaginary quadratic field of class number one. The class number is expressed as a finite sum in terms of the basic Jacobi elliptic functions, playing a similar role as the trigonometric sine in Dirichlet classical class number formula.

On biquadratic fields that admit unit power integral basis

We consider biquadratic number fields whose maximal orders have power integral bases consisting of units. We prove an effective and efficient criteria to decide whether the maximal order of a biquadratic field has a unit power integral basis or not. In particular we can determine all trivial biquadratic fields whose maximal orders have a unit power integral basis.

Deconstruction of a Dirichlet–Nazimow formula

Dirichlet (J Reine Angew Math 24, 1842) announced a generalization of his class number formula for real quadratic fields to biquadratic fields containing \({\mathbb{Q}(i)}\), by replacing the circular trigonometric functions with certain elliptic trigonometric functions. Subsequently, Nazimow (Ann Sci Ecole Norm Sup (3) 5:147–176, 1888) published the corresponding formula. Here we analyze the Dirichlet–Nazimow formula and update its proof in the context of Class Field Theory.

Hilbert genus fields of real biquadratic fields

Let \(K=\mathbb{Q}(\sqrt{p},\sqrt{d})\) be a real biquadratic field with p≡1 mod 4 or p=2, and d a squarefree positive integer. The Hilbert genus field is described explicitly by Yue (Ramanujan J. 21:17–25, 2010) in the case that p≡1 mod 4 and d≡3 mod 4. In this article, we give the Hilbert genus field of K explicitly for the remaining cases. We also consider the function field analogue of this problem.

Genus fields of real biquadratic fields

Let \(K={\mathbb{Q}}(\sqrt{p},\sqrt{d})\) be a real biquadratic field with prime p≡1 mod 4 and positive integer d≡3 mod 4. In this paper, we give the Hilbert genus field of K explicitly.

On the 2-Rank of the Hilbert Kernel of Number Fields

Abstract. Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the 2-rank of the Hilbert kernel of $E$ provided that the 2-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the 2-rank of $E$ whenever $F$ is totally real with trivial 2-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete 2-rank formula for the family of fields $\mathbb{Q}(\sqrt{2},\sqrt{\delta )}$ where $\delta $ is a squarefree integer.

Establishing the minimal index in a parametric family of bicyclic biquadratic fields

Let c ≥ 3 be a positive integer such that c, 4c + 1, c − 1 are square-free integers relatively prime in pairs. In this paper we find the minimal index and determine all elements with minimal index in the bicyclic biquadratic field K = ℚ(√(4c + 1)c, √(c − 1)c).

The additive unit structure of complex biquadratic fields

We determine which rings of the form Z(�) are generated by their units, whereis a root of the polynomial X4 BX2 + D such thatand all its conjugates are complex.

Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields

AbstractWe obtain Hauptmoduls of genus zero congruence subgroups of the type (p) := Γ0(p) + wp, where p is a prime and wp is the Atkin–Lehner involution. We then use the Hauptmoduls, along with modular functions on Γ1(p) to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

On congruence relations between the fundamental units of biquadratic fields

Given any distinct prime numbers p , q , and r satisfying certain simple congruence conditions, we display a congruence relation between the fundamental units for the biquadratic field K = Q ( p r , q ) , modulo a certain prime ideal of O K . This congruence in particular implies the validity of the equivariant Tamagawa number conjecture formulated by Burns and Flach for the pair ( h 0 ( Spec K ) , Z [ Gal ( K / Q ) ] ) .

On the 2-class field tower of some imaginary biquadratic number fields

Let \(k=Q({\sqrt d,\sqrt{-q}})\) be an imaginary biquadratic number field with Clk,2, the 2-class group of k, isomorphic to Z/2Z × Z/2mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1.

Index form equations in biquadratic fields: the $p$-adic case

Index form equations in biquadratic fields: the $p$-adic case

APPLICATION OF THE GENETIC ALGORITHM TO BLUME–EMERY–GRIFFITHS MODEL: TEST CASES

The equilibrium properties of the Blume–Emery–Griffiths (BEG) model Hamiltonian with the arbitrary bilinear (J), biquadratic (K) and crystal field interaction (D) are studied using the genetic algorithm technique. Results are compared with lowest approximation of the cluster variation method (CVM), which is identical to the mean field approximation. We found the genetic algorithm to be very efficient for fast search at the average fraction of the spins, especially in the early stages as the system is far from the equilibrium state. A combination of the genetic algorithm followed by one of the well-tested simulation techniques seems to be an optimal approach. The curvature of the inverse magnetic susceptibility is also presented for the stable state of the BEG model.

Un nouveau régulateur de type Gross

In this article we introduce a new regulator for arbitrary number fields, related to Gross’s regulator defined for CM-extensions. More precisely, this regulator is linked to the arithmetic of logarithmic classes, so that its non-triviality is equivalent to Gross’s generalized conjecture. As an application, we give a formula of Hasse type for the logarithmic class number of biquadratic fields containing the fourth root of unity.

Power integral bases in algebraic number fields whose Galois groups are 2-elementary abelian

Let K be a biquadratic field. M.-N. Gras and F. Tanoe gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [3]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFAesuaaa!40A2! $$\mathbb{Q}$$ whose Galois group is 2-elementary abelian. Then we shall prove that F of degree % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadA % eacaGG6aWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaa % cqWFAesucaGGDbGaeyyzImRaaGioaaaa!4673! $$[F:\mathbb{Q}] \geqq 8$$ , is monogenic if and only if % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 % da9mrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8Ng % HeLaaiikamaakaaabaGaeyOeI0IaaGymaaWcbeaakiaacYcadaGcaa % qaaiabgkHiTiaaikdaaSqabaGccaGGSaWaaOaaaeaacqGHsislcaaI % ZaaaleqaaOGaaiykaiabg2da9iab-PrirjaacIcacqaH2oGEdaWgaa % WcbaGaaGOmaiaaisdaaeqaaOGaaiykaaaa!5173! $$F = \mathbb{Q}(\sqrt { - 1} ,\sqrt { - 2} ,\sqrt { - 3} ) = \mathbb{Q}(\zeta _{24} )$$ under a suitable condition for the case of degree 8.

EQUILIBRIUM AND NONEQUILIBRIUM BEHAVIOR OF THE BLUME–EMERY–GRIFFITHS MODEL USING THE PAIR APPROXIMATION AND PATH PROBABILITY METHOD WITH THE PAIR DISTRIBUTION

Equilibrium properties of Blume–Emery–Griffiths (BEG) model with the arbitrary bilinear (J), biquadratic (K) and crystal field interaction (D) are studied on a body centered cubic lattice by using the pair approximation of the cluster variation method, which is identical to the Bethe approximation. The thermal variation of the stable, metastable and unstable dipole and quadrupolar moment order parameters are found for various values of the two different coupling parameters J/K and D/K. The phase transitions of the stable, metastable and unstable branches of the order parameters are investigated extensively. The critical temperatures in the case of a second-order phase transition are obtained for different values of J/K and D/K by the Hessian determinant. The first-order phase transition temperature is determined by matching the values of the two branches of the free energy values. On the other hand, nonequilibrium behavior of the model is studied by using the path probability method with the pair distribution and the set of nonlinear differential equations, which is also called the dynamic or rate equations, is obtained. The solution of the dynamic equations is expressed by means of flow diagrams. The stable, metastable and unstable solutions are shown and the "overshooting" phenomenon is seen in the flow diagrams, explicitly. The role of the unstable points, as separators between the stable and the metastable points, is described and how a system freezes in a metastable state is also investigated, extensively. Moreover, the dynamic equations are also solved by using the Runge–Kutta method in order to study the relaxation of order parameters. In this investigation, the "flatness" property of metastable states and the "overshooting" phenomenon are seen explicitly.