Unramified extensions of quadratic number fields with Galois group 2.An

In this paper, we provide infinite families of quadratic number fields with everywhere unramified Galois extensions of Galois group [Formula: see text] and [Formula: see text], respectively. To my knowledge, these are the first instances of infinitely many such realizations for perfect groups which are not generated by involutions, a property which makes them difficult to approach for the problem in question and leads to somewhat delicate local–global problems in inverse Galois theory.

Let p and q be two positive primes, let \(\ell\) be an odd positive prime and let F be a quadratic number field. Let K be an extension of F of degree \(\ell\) such that K is a dihedral extension of \({\mathbb {Q}}\), or else let K be an abelian \(\ell\)-extension of F unramified over F whenever \(\ell\) divides the class number of F. In this paper, we provide a complete characterization of division quaternion algebras \(H_{K}(p, q)\) over K.

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-$p$extensions of imaginary quadratic number fields for$p$an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of$\mathbb{F}_{q}(t)$, the Galois groups of the maximal unramified pro-$p$extensions, as$q\rightarrow \infty$, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

Erratum: Unramified extensions of quadratic number fields with certain perfect Galois groups

We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ‘arithmetic Bertini theorem’ conjectured by Poonen for ${\mathbb {P}}^1_{\mathbb {Z}}$ . Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$ number fields of degree n having associated Galois group $S_n$ and absolute discriminant less than X, improving the best previously known lower bound of $\gg X^{1/2+1/n}$ . Finally, our methods correct an error in and thus resurrect earlier (retracted) results of Nakagawa on lower bounds for the number of totally unramified $A_n$ -extensions of quadratic number fields of bounded discriminant.

Unramified extensions of quadratic number fields, I

Unramified extensions of quadratic number fields, II

Artin’s Reciprocity Law (1923/26) for Abelian extensions of algebraic number fields is the central theorem of Class Field Theory which, by the Theory of Takagi (1920) [64], is the Theory of Abelian Extensions of Algebraic Number Fields. Abelian extensions of algebraic number fields have been studied extensively in the second half of the 19-th century,in particular by Kronecker(1853,1856,1877,1882), Weber (1886/87, 1897/98) and Hilbert (1896, 1897, 1898) who laid the foundations and discovered many fundamental properties of the Class Fields. These discoveries were made possible by the thorough study of a particular kind of Abelian extensions, namely the study of cyclotomic fields initiated by Gauss (1801) [24, Sect. 7] and carried further by Kummer (1847-1874). The term Abelian in connection with algebraic extensions (or, at that time, algebraic equations) was coined by Kronecker in 1853, first related to algebraic equations as ‘Abelian equation.’ 1 By this Kronecker referred to polynomials with cyclic Galois group. However, Kronecker was already then aware of the more general polynomials with Abelian (in the modern sense) Galois group, since he is referring on page 6 (in [45], Werke, Bd. 4) to the fundamental treatise of Abel of 1829 which appeared in Volume 4 of Crelle’s Journal [1], just two months before Abel’s death, and where Abel states explicitly the condition for commutativity θθ 1x =θ1θx in his assertion that a polynomial with Abelian Galois group is solvable (with radicals). Abel says there on page 479 of [1]: “En general j’ai demontre le theoreme suivant: ‘Si les racines d’une equation d’un degre quelconque sont liees entre elles de telle sorte, que toutes ces racines puissent etre exprimees rationnellement au moyen de l’une d’elles, que nous d’signerons par x; si de plus, en designant par θx, θ1 x deux autres racines quelconques, on a θ1 x = θθ 1 6x, 1’ equation dont il s’agit sera toujours resoluble algebriquement.’ ” (In general I have demonstrated the following theorem: If the roots of an equation of any degree are related to each other in such a way that all these roots can be expressed rationally by means of one of them, which will be denoted by x; if, in addition, we have θθ1x = θ1θx, where we denote by θx, θ1 x any two other roots, then the equation in question is algebraically solvable [that is, solvable by radicals].) Later in 1877 [48] Kronecker used the term ‘Abelian equation’in the larger sense to mean a polynomial with Abelian Galois group in our modern sense. He says there on page 66 (in [48], Werke, Bd. 4) that he now calls ‘Abelian equation’ an equation having the property that all its roots x are rational functions of any one of them and if θ 1 and θ2 are two of these functions then θ1θ2x = θ2θ1x. He then calls ‘simple Abelian equation’ the equation he treated in 1853, namely the equation with cyclic Galois group of prime order. That he had already used the term ‘Abelian equation’ in the paper of 1853 in the special case of cyclic equations is justified by Kronecker by the fact that the ‘Abelian equation’ can be reduced to the ‘simple Abelian equation,’ or as Kronecker says on page 69 (in [48], Werke, Bd. 4), that every root of any Abelian equation is a rational function of roots of simple Abelian equations.2 This justification is repeated in his paper of 1882 on the composition of Abelian equations.3 In the paper of 1853 Kronecker also states the famous theorem that any root of a (simple) Abelian equation [that is a cyclic equation] with integral rational coefficients can be represented by roots of unity, that is, is contained in a cyclotomic field over the rational number field.4 The same theorem is stated by Kronecker in his paper of 1877 in the case of the general Abelian equation with integral rational coefficients.5 It was later proved by Weber (1886) [67] and more simply by Hilbert (1896) [41]. In both papers Kronecker also stated his Jugendtraum for an analogous theorem for Abelian equations with coefficients in a quadratic imaginary number field. This theorem was proved partially by Weber (1908) [69] and Fueter (1914) [23] and completely by Takagi (1920) [64]. Both theorems have plaid a crucial role in the history of class field theory.6

We 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.

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].

Letkbe an algebraic number field andCkits ideal class group (in the wider sense). SupposeKis a finite extension ofk. Then we say that an ideal class ofk capitulatesinKif this class is in the kernel of the homomorphisminduced by extension of ideals fromktoK(See Section 2 below). In [4], Iwasawa gives examples of real quadratic number fields,with distinct primesPi≡ 1 (mod 4), for which all the ideal classes of the 2-class group,Ck,2(the 2-Sylow subgroup ofCk), capitulate in an unramified quadratic extension ofk. In these examples,Ck,2is abelian of type (2,2), i.e. isomorphic to ℤ/2ℤ×ℤ/2ℤ and so all four ideal classes capitulate.

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.

In the previous paper [15], we determined the structure of the Galois groups Gal(K ur /K) of the maximal unramified extensions K ur of imaginary quadratic number fields K of conductors ≦1000 under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors ≧723) and give a table of Gal(K ur /K). We update the table (under GRH). For 19 exceptional fields K of them, we determine Gal(K ur /K). In particular, for K=𝐐(-856), we obtain Gal(K ur /K)≅S 4 ˜×C 5 andK ur =K 4 , the fourth Hilbert class field of K. This is the first example of a number field whose class field tower has length four.

We give a construction of unramified cyclic octic extensions of certain complex quadratic number fields. The binary quadratic form used in this construction also shows up in the theory of 2-descents on Pell conics and elliptic curves, as well as in the explicit description of cyclic quartic extensions.