THE FERMAT QUARTIC X 4 + Y 4 = 2 m IN QUADRATIC NUMBER FIELDS

Let K be a number field and $$K_\mathrm{ur}$$ be the maximal extension of K that is unramified at all places. In a previous article (Kim, J Number Theory 166:235–249, 2016), the first author found three real quadratic fields K such that $$\mathrm {Gal}(K_\mathrm{ur}/K)$$ is finite and non-abelian simple under the assumption of the generalized Riemann hypothesis (GRH). In this article, we extend the methods of Kim (2016) and identify more quadratic number fields K such that $$\mathrm {Gal}(K_\mathrm{ur}/K)$$ is a finite nonsolvable group and also explicitly calculate their Galois groups under the assumption of the GRH. In particular, we find the first imaginary quadratic field with this property.

This article presents algorithms for computing discrete logarithms in class groups of quadratic number fields. In the case of imaginary quadratic fields, the algorithm is based on methods applied by Hafner and McCurley [HM89] to determine the structure of the class group of imaginary quadratic fields. In the case of real quadratic fields, the algorithm of Buchmann [Buc89] for computation of class group and regulator forms the basis. We employ the rigorous elliptic curve factorization algorithm of Pomerance [Pom87], and an algorithm for solving systems of linear Diophantine equations proposed and analysed by Mulders and Storjohann [MS99]. Under the assumption of the Generalized Riemann Hypothesis, we obtain for fields with discriminant d a rigorously proven time bound of \(L_{|d|} [\frac{1}{2}, \frac{3}{4}\sqrt{2}]\).

We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.

Let K be an algebraic number field of finite degree over the rational field Q, and aK(n) the number of integral ideals in K with norm n. When K is a Galois extension over Q, many authors contribute to the integral power sums of aK(n), $$\sum\limits_{n \leqslant x} {a\kappa {{\left( n \right)}^l}} ,\;l = 1,\;2,\;3, \ldots $$ . This paper is interested in the distribution of integral ideals concerning different number fields. The author is able to establish asymptotic formulae for the convolution sum $${\sum\limits_{n \leqslant x} {a{\kappa _1}{{\left( {{n^j}} \right)}^l}a{\kappa _2}\left( {{n^j}} \right)} ^l},\;\;j = 1,\;2,\;\;l = \;2,\;3, \ldots $$ , where K1 and K2 are two different quadratic fields.

The distribution of $$\alpha p$$ modulo one, where p runs over the rational primes and $$\alpha $$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $$\nu >0$$ one can establish the infinitude of primes p satisfying $$||\alpha p||\le p^{-\nu }$$ . The latest record in this regard is Kaisa Matomäki’s landmark result $$\nu =1/3-\varepsilon $$ which presents the limit of currently known technology. Recently, Glyn Harman, and, jointly, Marc Technau and the first-named author, investigated the same problem in the context of imaginary quadratic fields. Glyn Harman obtained an analog for $$\mathbb {Q}(i)$$ of his result in the context of $$\mathbb {Q}$$ , which yields an exponent of $$\nu =7/22$$ . Marc Technau and the first-named author produced an analogue of Bob Vaughan’s result $$\nu =1/4-\varepsilon $$ for all imaginary quadratic number fields of class number 1. In the present article, we establish an analog of the last-mentioned result for real quadratic fields of class number 1 under a certain Diophantine restriction. This setting involves the additional complication of an infinite group of units in the ring of integers. Moreover, although the basic sieve approach remains the same (we use an ideal version of Harman’s sieve), the problem takes a different flavor since it becomes truly 2-dimensional. We reduce it eventually to a counting problem which is, interestingly, related to roots of quadratic congruences. To approximate them, we use an approach by Christopher Hooley based on the theory of binary quadratic forms.

Dyadic ideal, class group, and tame kernel in quadratic fields

We provide several criteria to show over which quadratic number fields \mathbb{Q}(\sqrt{D}) there is a nonconstant arithmetic progression of five squares. This is carried out by translating the problem to the determination of when some genus five curves C_D defined over \mathbb{Q} have rational points, and then by using a Mordell–Weil sieve argument. Using an elliptic curve Chabauty-like method, we prove that, up to equivalence, the only nonconstant arithmetic progression of five squares over \mathbb Q(\sqrt{409}) is 7^2 , 13^2 , 17^2 , 409 , 23^2 . Furthermore, we provide an algorithm for constructing all the nonconstant arithmetic progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.

There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\), there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\).

Let $n$ be an integer. Then, it is well known that there are infinitely many\nimaginary quadratic fields with an ideal class group having a subgroup isomorphic to\n$\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$. Less is known for real\nquadratic fields, other than the cases that $n=3,5,$ or $7$, due to Craig [3] and\nMestre [4, 5]. In this article, we will prove that there exist infinitely many real\nquadratic number fields with the ideal class group having a subgroup isomorphic to\n$\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$ In addition, we will prove\nthat there exist infinitely many imaginary quadratic number fields with the ideal\nclass group having a subgroup isomorphic to $\\mathbb{Z}/n\\mathbb{Z} \\times\n\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$.

Let K be a number field and let G be a finite abelian group. We call K a Hilbert-Speiser field of type G if, and only if, every tamely ramified normal extension L/K with Galois group isomorphic to G has a normal integral basis. Now let C 2 and C 3 denote the cyclic groups of order 2 and 3, respectively. Firstly, we show that among all imaginary quadratic fields, there are exactly three Hilbert-Speiser fields of type $C_{2}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-1, -3, -7\}$. Secondly, we give some necessary and sufficient conditions for a real quadratic field $K = \mathbb{Q}(\sqrt {m})$ to be a Hilbert-Speiser field of type C 2. These conditions are in terms of the congruence class of m modulo 4 or 8, the fundamental unit of K, and the class number of K. Finally, we show that among all quadratic number fields, there are exactly eight Hilbert-Speiser fields of type $C_{3}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}$.

The purpose of this article is to describe simple ways to construct quadratic number fields each having an unramified extension which properly contains the Hilbert class field of its genus field (in the wide sense). The motivation of this study is the author’s observation that under the Generalized Riemann Hypothesis (GRH), for most quadratic number fields of small conductors, their maximal unramified extensions coincide with the Hilbert class fields of their genus fields. More precisely, under GRH, among the 305 imaginary quadratic number fields with discriminants larger than —1000, at most 16 fields are exceptional [39], [40], and among the 1690 real quadratic number fields with discriminants less than or equal to 5565, only 4 fields are exceptional [41].

This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between elliptic units and class numbers.

Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one. The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ -function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h ( d ) formula in real quadratic fields claims that we have h ( d ) . log ε d = Δ . ℒ ( 1 , χ d ) h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d > 0 and the fundamental unit ɛ d of ℚ ( d ) {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ -function and fundamental unit ɛ d are significant and necessary tools for determining the structure of real quadratic fields. The focus of this paper is to determine structure of some special real quadratic fields for d > 0 and d ≡ 2,3 ( mod 4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element w d , fundamental unit ɛ d , and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

Infrastructures are group-like objects that make their appearance in arithmetic geometry in the study of computational problems related to number fields and function fields over finite fields. The most prominent computational tasks of infrastructures are the computation of the circumference of the infrastructure and the generalized discrete logarithms. Both these problems are not known to have efficient classical algorithms for an arbitrary infrastructure. Our main contributions are polynomial time quantum algorithms for one-dimensional infrastructures that satisfy certain conditions. For instance, these conditions are always fulfilled for infrastructures obtained from number fields and function fields, both of unit rank one. Since quadratic number fields give rise to such infrastructures, this algorithm can be used to solve Pell's equation and the principal ideal problem. In this sense we generalize Hallgren's quantum algorithms for quadratic number fields, while also providing a polynomial speedup over them. Our more general approach shows that these quantum algorithms can also be applied to infrastructures obtained from complex cubic and totally complex quartic number fields. Our improved way of analyzing the performance makes it possible to show that these algorithms succeed with constant probability independent of the problem size. In contrast, the lower bound on the success probability due to Hallgren decreases as the fourth power of the logarithm of the circumference. Our analysis also shows that fewer qubits are required. We also contribute to the study of infrastructures, and show how to compute efficiently within infrastructures.

It has recently been established that there are exactly seven Witt equivalence classes of quadratic number fields, and then all quadratic and cubic number fields have been classified with respect to Witt equivalence. In this paper we have classified number fields of degree four. Using this classification, we have proved the Conjecture of Szymiczek about the representability of Witt equivalence classes by quadratic extensions of quadratic fields.