Quadratic Subfield Research Articles

On quadratic subfields of generalized quaternion extensions

We give necessary and sufficient conditions for the embedding of a quadratic extension of a number field $k$ into an extension with group of generalized quaternions; in this case, the case of both a cyclic kernel and a generalized quaternion is considered. As a consequence, it is proved that the class of ultrasolvable $2$-extensions with cyclic kernel does not coincide with the class of non-semidirect extensions. Sufficient conditions are also given for the embedding of quadratic extensions $k(\sqrt{d_1})/k$, $k(\sqrt{d_2})/k$, $k(\sqrt{d_1d_2})/k$ of a number field $k$ into a generalized quaternion extension $L/k$. Related examples are given.

On quadratic subfields of generalized quaternion extensions

Мы даем необходимые и достаточные условия погружения квадратичного расширения числового поля $k$ в расширение с группой обобщенных кватернионов; при этом рассматривается случай как циклического ядра, так и обобщенно-кватернионного. Как следствие, доказывается, что класс ультраразрешимых $2$-расширений с циклическим ядром не совпадает с классом неполупрямых расширений. Также даются достаточные условия погружения квадратичных расширений $k(\sqrt{d_1})/k$, $k(\sqrt{d_2})/k$, $k(\sqrt{d_1d_2})/k$ числового поля $k$ в обобщенно-кватернионное расширение $L/k$. Рассматриваются примеры. Библиография: 14 наименований.

Algebraic number fields generated by an infinite family of monogenic trinomials

For an infinite family of monogenic trinomials \(P(X)=X^3\pm 3rbX-b\in {\mathbb {Z}}[X]\), arithmetical invariants of the cubic number field \(L={\mathbb {Q}}(\theta )\), generated by a zero \(\theta\) of \(P(X)\), and of its Galois closure \(N=L(\sqrt{d_L})\) are determined. The conductor \(f\) of the cyclic cubic relative extension \(N/K\), where \(K={\mathbb {Q}}(\sqrt{d_L})\) denotes the unique quadratic subfield of \(N\), is proved to be of the form \(3^eb\) with \(e\in \lbrace 1,2\rbrace\), which admits statements concerning primitive ambiguous principal ideals, lattice minima, and independent units in \(L\). The number \(m\) of non-isomorphic cubic fields \(L_1,\ldots ,L_m\) sharing a common discriminant \(d_{L_i}=d_L\) with \(L\) is determined.

Smallness of Faltings heights of CM abelian varieties

We prove that assuming the Colmez conjecture and the “no Siegel zeros” conjecture, the stable Faltings height of a CM abelian variety over a number field is less than or equal to the logarithm of the root discriminant of the field of definition of the abelian variety times an effective constant depending only on the dimension of the abelian variety. In view of the fact that the Colmez conjecture for abelian CM fields, the averaged Colmez conjecture, and the “no Siegel zeros” conjecture for CM fields with no complex quadratic subfields are already proved, we prove unconditional analogues of the result above. In addition, we also prove that the logarithm of the root discriminant of the field of everywhere good reduction of CM abelian varieties can be “small”.

Triquadratic p-rational fields

In his work about Galois representations, R. Greenberg conjectured the existence, for any odd prime p and any positive integer t, of a multiquadratic p-rational number field of degree 2t.In this article, we prove that there exists infinitely many primes p such that the triquadratic field Q(p(p+2),p(p−2),−1) is p-rational.To do this, we use an analytic result providing us with infinitely many primes p such that p−2 and p+2 simultaneously have large square factors. Therefore the related imaginary quadratic subfields Q(−p(p+2)), Q(−p(p−2)) and Q(−(p+2)(p−2)) have small discriminants for infinitely many primes p. In the spirit of Brauer-Siegel estimates, it proves that the class numbers of these imaginary quadratic fields are relatively prime to p, and so prove their p-rationality.

Quaternion algebras with derivations

We study derivations on quaternion algebras that stabilise quadratic subfields. Following the work of L. Juan and A. Magid [10] , we provide an explicit construction of a differential splitting field for a given differential quaternion algebra. We also examine the presence and impact of those derivations on quaternion algebras that admit new constants.

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.

The number of $D_4$-fields ordered by conductor

We consider families of quartic number fields whose normal closures over \mathbb{Q} have Galois group isomorphic to D_4 , the symmetries of a square. To any such field L , one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image D_4 . We determine the asymptotic number of such D_4 -quartic fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order-4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L -function methods. Both of these strategies fail in the case of D_4 -quartic fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of D_4 combined with both approaches mentioned above to obtain exact asymptotics.

On the 2-rank and 4-rank of the class group of some real pure quartic number fields

Abstract Let K = ℚ ( p d 2 4 ) $\begin{array}{} \displaystyle (\sqrt[4]{pd^{2}}) \end{array}$ be a real pure quartic number field and k = ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ) its real quadratic subfield, where p ≡ 5 (mod 8) is a prime integer and d an odd square-free integer coprime to p. In this work, we calculate r 2(K), the 2-rank of the class group of K, in terms of the number of prime divisors of d that decompose or remain inert in ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ), then we will deduce forms of d satisfying r 2(K) = 2. In the last case, the 4-rank of the class group of K is given too.

The 2-rank of the class group of some real cyclic quartic number fields

In this paper, we investigate the 2-rank of the class group of some real cyclic quartic number fields. Precisely, we consider the case where the quadratic subfield is \(\mathbb {Q}(\sqrt{\ell })\) with \(\ell \equiv 5(\mathrm{mod}\ 8)\) is a prime.

Certain properties of the tensor product of two Albert forms

Let F be a field, Let further D1, D2 be biquaternion algebras over F with Albert forms We prove that the dimension of anisotropic part of the form is at most 24, provided there exists a common splitting field of degree for D1 and D2. We also investigate the symbol length of i.e., the minimal number n such that is the sum of n 4-fold Pfister forms over F modulo It turns out that if is a biquaternion algebra as well, then and this inequality is strict. Among negative results we give examples of biquaternion division algebras D1, D2 without common quadratic subfield, and such that is a biquaternion algebra.

Primary rank of the class group of real cyclotomic fields

Let H=ℚ(ζn+ζn−1) and ℓ be an odd prime such that q≡1(modℓ) for a prime factor q of n. We get a bound on the ℓ-rank of the class group of H in terms of the ℓ-rank of the class group of a real quadratic subfield contained in H. At the end we look into few numerical examples.

On the Prime Geodesic Theorem for SL4

In 1949, A. Selberg discovered a real variable (an elementary) proof of the prime number theorem. A number of authors have adapted Selberg’s method to achieve quite a good corresponding error term. The Riemann hypothesis has never been proved or disproved however. Any generalization of the prime number theorem to the more general situations is known in literature as a prime geodesic theorem. In this paper we derive yet another proof of the prime geodesic theorem for compact symmetric spaces formed as quotients of the Lie group SL4 (R). While the first known proof in this setting applies contour integration over square boundaries, our proof relies on an application of modified circular boundaries. Recently, A. Deitmar and M. Pavey applied such prime geodesic theorem to derive an asymptotic formula for class numbers of orders in totally complex quartic fields with no real quadratic subfields.

Construction of class fields over imaginary biquadratic fields

Let K be an imaginary biquadratic field, K1, K2 be its imaginary quadratic subfields and K3 be its real quadratic subfield. For integers N > 0, µ ≥ 0 and an odd prime p with gcd(N,p) = 1, let K(Npµ) and (Ki)(Npµ) for i = 1,2,3 be the ray class fields of K and Ki, respectively, modulo Np µ . We first present certain class fields ^ K 1,2 N,p,µ of K, in the sense of Hilbert, which are generated by ray class invariants

PólyaS3-extensions of ℚ

AbstractA number fieldKwith a ring of integers 𝒪Kis called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪Khas a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to generalS3-extensions of ℚ. Also, we prove for a real (resp. imaginary) PólyaS3-extensionLof ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield ofL, we determine when these sharp upper bounds can occur.

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.

Number fields with large minimal index containing quadratic subfields

In this paper, we consider number fields containing quadratic subfields with minimal index that is large relative to the discriminant of the number field. We give new upper bounds on the minimal index, and construct families with the largest possible minimal index.

On the unit index of some real biquadratic number fields

Let $p_1\equiv p_2\equiv1 \pmod 4$ be different prime numbers such that $\left(\dfrac{2}{p_2}\right)=\left(\dfrac{p_1}{p_2}\right)=-\left(\dfrac{2}{p_1}\right)=-1$. Put $\kk=\QQ(\sqrt{2p_1p_2})$ and let $\KK$ be a quadratic extension of $\kk$ contained in its absolute genus field $\kk^{(*)}$. Denote by $k_j$, $1\leq j\leq 3$, the three quadratic subfields of $\KK$. Let $E_{\KK}$ (resp. $E_{k_j}$) be the unit group of $\KK$ (resp. $k_j$). The unit index $\left[E_{\KK}: \prod_{j=1}^3E_{k_j}\right]$ is characterized in terms of biquadratic residue symbols between $2$, $p_1$ and $p_2$ or by the capitulation of $\mathfrak{2}$, the prime ideal of $\QQ(\sqrt{2p_1})$ above $2$, in $\KK$. These results are used to describe the $2$-rank of some CM-fields.

Remarks on automorphy of residually dihedral representations

We prove automorphy lifting results for geometric representations $\rho:G_F \rightarrow GL_2(\mathcal{O})$, with $F$ a totally real field, and $\mathcal{O}$ the ring of integers of a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime, such that the residual representation $\bar{\rho}$ is totally odd and induced from a character of the absolute Galois group of the quadratic subfield $K$ of $F(\zeta_p)/F$. Such representations fail the Taylor-Wiles hypothesis and the patching techniques to prove automorphy do not work. We apply this to automorphy of elliptic curves $E$ over $F$, when $E$ has no $F$ rational 7-isogeny and such that the image of $G_F$ acting on $E[7]$ normalizes a split Cartan subgroup of $GL_2(\mathbb{F}_7)$.

Hilbert Genus Fields of Imaginary Biquadratic Fields

Let $$K_0={\mathbb {Q}}\left( \sqrt{\delta }\right) $$ be a quadratic field. For those $$K_0$$ with odd class number, much work has been done on the explicit construction of the Hilbert genus field of a biquadratic extension $$K={\mathbb {Q}}\left( \sqrt{\delta },\sqrt{d}\right) $$ over $${\mathbb {Q}}$$ . When $$\delta =2$$ or p with $$p\equiv 1\bmod 4$$ a prime and K is real, it was described in Yue (Ramanujan J 21:17–25, 2010) and Bae and Yue (Ramanujan J 24:161–181, 2011). In this paper, we describe the Hilbert genus field of K explicitly when $$K_0$$ is real and K is imaginary. In fact, we give the explicit construction of the Hilbert genus field of any imaginary biquadratic field which contains a real quadratic subfield of odd class number.