Biquadratic Number Fields Research Articles

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.

On the rank of the 2-class group of some imaginary biquadratic number fields

On the rank of the 2-class group of some imaginary biquadratic number fields

An Infinite Family of Number Fields with No Inert Primes

The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk is the Dedekind domain of the integer elements of K. To prove such a result, consider for any prime p, the decomposition into a product of prime ideals of Zk, of the ideal . From this point, we use on the one hand: 1) The well- known property that says: If , then the ideal pZk decomposes into a product of prime ideals of Zk as following: . (where: ; is the irreducible polynomial of θ, and, is its reduction modulo p, which leads to a product of irreducible polynomials in Fp[X]). It is clear that because if is reducible in Fp[X], then consequently p is not inert. Now, we prove the existence of such p, by proving explicit such p as follows. So we use on the other hand: 2) this property that we prove, and which is: If , is an irreducible normalized integer polynomial, whose splitting field is , then for any prime number p ∈ N: is always a reducible polynomial. 3) Consequently, and this closes our proof: let’s consider the set (whose cardinality is infinite) of monogenic biquadratic number fields: . Then each fθ(X) checks the above properties, this means that for family M, all its fields, do not admit any inert prime numbers p ∈ N. 2020-Mathematics Subject Classification (MSC2020) 11A41 - 11A51 - 11D25 - 11R04 - 11R09 - 11R11 - 11R16 - 11R32 - 11T06 - 12E05 - 12F05 -12F10 -13A05-13A15 - 13B02 - 13B05 - 13B10 - 13B25 -13F05

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].

English

Let $$\mathbb{k} = \mathbb{Q} \left( {\sqrt 2 ,\; \sqrt d } \right)$$ be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and $$\mathbb{k}_2^{\left( 2 \right)}$$ its second Hilbert 2-class field. Denote by $$G = {\rm{Gal}}\left( {\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}} \right)$$ the Galois group of $${\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}}$$ . The purpose of this note is to investigate the Hilbert 2-class field tower of $$\mathbb{k}$$ and then deduce the structure of G.

Minimal index of bicyclic biquadratic number fields

T. Nakahara proved that the set of minimal indices of bicyclic biquadratic number fields with field index 1 is unbounded. We strengthen his result by showing that this set just coincides with the set of positive integers, and every positive integer occurs infinitely many times as minimal index of totally complex bicyclic biquadratic number fields. Moreover, we study the analogous problem for all the other possible values of the field index.

Biquadratic fields having a non-principal Euclidean ideal class

H.W. Lenstra [7] introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in [2] and [6]). In this paper, we construct certain new families of biquadratic number fields having a non-principal Euclidean ideal class and this extends the previously known families given by H. Graves [2] and C. Hsu [6].

Strongly modular models of ℚ-curves

Let [Formula: see text] be a [Formula: see text]-curve without complex multiplication. We address the problem of deciding whether [Formula: see text] is geometrically isomorphic to a strongly modular [Formula: see text]-curve. We show that the question has a positive answer if and only if [Formula: see text] has a model that is completely defined over an abelian number field. Next, if [Formula: see text] is completely defined over a quadratic or biquadratic number field [Formula: see text], we classify all strongly modular twists of [Formula: see text] over [Formula: see text] in terms of the arithmetic of [Formula: see text]. Moreover, we show how to determine which of these twists come, up to isogeny, from a subfield of [Formula: see text].

Universal quadratic forms and indecomposables over biquadratic fields

AbstractThe aim of this article is to study (additively) indecomposable algebraic integers of biquadratic number fields K and universal totally positive quadratic forms with coefficients in . There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field K. Furthermore, estimates are proven which enable algorithmization of the method of escalation over K. These are used to prove, over two particular biquadratic number fields and , a lower bound on the number of variables of a universal quadratic forms.

The extended Bloch groups of biquadratic and dihedral number fields

In this paper, we study the Galois action on the extended Bloch groups of biquadratic and dihedral number fields. We prove that if F is a biquadratic number field, then the index Q2(F) in Browkin and Gangl's formulas on the Brauer–Kuroda relation can only be 1 or 2. This is exactly what Browkin and Gangl predicted in their paper. Moreover we give the explicit criteria for Q2(F)=1 or 2 in terms of the Tate kernels. We also prove that Q2(F)=1 or p for any dihedral extension F/Q whose Galois group is the dihedral group of order 2p, where p is an odd prime.

Elements with prime and small indices in bicyclic biquadratic number fields

We give necessary and sufficient conditions for the existence of primitive algebraic integers with index A in totally complex bicyclic biquadratic number fields where A is an odd prime or a positive rational integer at most 10. We also determine all these elements and prove that there are infinitely many totally complex bicyclic biquadratic number fields containing elements with index A.

Some non-Pólya biquadratic fields with low ramification

Pólya fields are fields with principal Bhargava factorial ideals, and as a generalization of class number one number fields, their classification might be of interest to number theorists. It is known that Pólya fields have little ramification, and the aim of this paper is to prove non-Pólyaness of an infinite family of biquadratic number fields with 3 or 4 primes of ramification, correcting a minor mistake in the literature. It turns out that finer arithmetic invariants of the field such as the Hasse unit index plays a direct role in some cases.

Pólya groups of the imaginary bicyclic biquadratic number fields

Let OK and CK 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 OK with norm q, if these ideals do not exist we set ∏q(K)=OK. The Pólya group of K is the subgroup of CK generated by the classes of the ideals ∏q(K), and K is called a Pólya field if the module of integer-valued polynomials over OK 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.

Capitulation in the Absolutely Abelian Extensions of some Number Fields II

We study the capitulation of 2-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields $\mathbb {k} =\mathbb {Q}(\sqrt {pq_{1}q_{2}}, i)$ , where $i=\sqrt {-1}$ and q 1≡q 2≡−p≡−1 (mod 4) are different primes. For each of the three quadratic extensions $\mathbb {K}/\mathbb {k}$ inside the absolute genus field 𝕜 (∗) of 𝕜, we compute the capitulation kernel of $\mathbb {K}/\mathbb {k}$ . Then we deduce that each strongly ambiguous class of $\mathbb {k}/\mathbb {Q}(i)$ capitulates already in 𝕜 (∗).

Restriction of Coherent Hilbert Eisenstein series

In (Pure Appl Math Q 1:305–340, 2005), Yang constructed a family of coherent Hilbert Eisenstein series and asked several questions concerning the span of their diagonal restrictions. In this paper, we will answer one of them, where the construction essentially comes from a biquadratic number field K. Surprisingly, the span depends on the non-vanishing of the central critical value of the L-function associated to a weight 2 eigenform.

English

We study the capitulation of ideal classes in an infinite family of imaginary bicyclic biquadratic number fields consisting of fields k = Q( √ 2pq, i), where i = √ −1 and p ≡ −q ≡ 1 (mod 4) are different primes. For each of the three quadratic extensions K/k inside the absolute genus field k(∗) of k, we compute the capitulation kernel of K/k. Then we deduce that each strongly ambiguous class of k/Q(i) capitulates already in k(∗), which is smaller than the relative genus field (k/Q(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(*).

On the Hilbert 2-class field tower of some abelian 2-extensions over the field of rational numbers

It is well known by results of Golod and Shafarevich that the Hilbert 2-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian 2-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian 2-extension over ℚ in which eight primes ramify and one of theses primes ≡ −1 (mod 4), the Hilbert 2-class field tower is infinite.

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.

Generating a power basis over a Dedekind ring

Let R be a Dedekind ring, K its quotient field, L a separable finite extension over K , and O L the integral closure of R in L . In this paper we provide a “practical” criterion that tests when a given α ∈ O L generates a power basis for O L over R (i.e. when O L = R [ α ] ), improving significantly a result in this direction by M. Charkani and O. Lahlou. Applications in the context of cyclotomic, quadratic, biquadratic number fields, and some Dedekind rings are provided.