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Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x, Comment: Corrected typos

We determine those imaginary quadratic number fields whose 2-class groups have rank 3 and 4-rank ≤2 and such that the 2-class groups of their Hilbert 2-class fields have rank 2. This result, along with previous work, gives a complete determination of all complex quadratic number fields for which the 2-class groups of their 2-class fields have rank at most 2.

A family $\mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}\in \mathcal{F}$ can be made uniformly bounded after removing from $\mathcal{F}$ those whose number field degrees lie in a subset of $\mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $\mathcal{E}_F$ of elliptic curves defined over number fields and with $F$-rational $j$-invariant is typically bounded in torsion. For any integer $d\in\mathbb{Z}^+$, we also strengthen a result on typically bounding torsion for the family $\mathcal{E}_d$ of elliptic curves defined over number fields and with degree $d$ $j$-invariant., 17 pages, to appear in Journal of Number Theory

Let K / F be a finite Galois extension of number fields and let σ be an absolutely irreducible, self-dual, complex valued representation of Gal ( K / F ) . Let p be an odd prime and consider two elliptic curves E 1 , E 2 defined over Q with good, ordinary reduction at primes above p and equivalent mod-p Galois representations. In this article, we study the variation of the parity of the multiplicities of σ in the representation space associated to the p ∞ -Selmer groups of E 1 and E 2 over K. We also compare the root numbers for the twists of E 1 and E 2 over F by σ and show that the p-parity conjecture holds for the twist of E 1 / F by σ if and only if it holds for the twist of E 2 / F by σ. We also express Mazur-Rubin-Nekovař's arithmetic local constants in terms of certain local Iwasawa invariants.

We develop efficient algorithms for determining the number of global cubic fields of given discriminant. They are based on earlier work of Hasse for cubic number fields. Our new ingredients allow to enlarge the range of computations considerably. For explicit calculations we used Magma.

Let f n ( k ) be the number of subrings of index k in Z n . We show that results of Brakenhoff imply a lower bound for the asymptotic growth of subrings in Z n , improving upon lower bounds given by Kaplan, Marcinek, and Takloo-Bighash. Further, we prove two new lower bounds for f n ( p e ) when e ≥ n − 1 . Using these bounds, we study the divergence of the subring zeta function of Z n and its local factors. Lastly, we apply these results to the problem of counting orders in a number field.

The p-rationality of a totally real abelian number field can be checked from the values L ( 2 − p , χ ) of Dirichlet L-functions, for all non-principal even Dirichlet characters associated to the field. Using this criterion and the properties of the generalized Bernoulli numbers, we study the p-rationality of Q ( ζ 2 l + 1 ) + , the maximal real subfield of Q ( ζ 2 l + 1 ) , for Sophie Germain primes l and odd primes p that are primitive roots modulo l. We prove that Q ( ζ 2 l + 1 ) + is p-rational for such pairs if p 4 l . We also prove that the Siegel's heuristics on the equidistribution of the residues of Bernoulli numbers modulo p imply that Q ( ζ 2 l + 1 ) + is p-rational for all but finitely many p that are primitive roots modulo l.

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

A number field is called a Polya field if the module of integer-valued polynomials over its ring of integers has a regular basis. Let L be a field which is a compositum of two quadratic Polya fields. Some questions were raised on Polyaness of L in [7] . Part was solved in [3] and [8] . Here we develop a general strategy allowing us to treat the remaining cases but also to find all these previous results.

We construct higher-dimensional Calabi–Yau varieties defined over a given number field with Zariski dense sets of rational points. We give two elementary constructions in arbitrary dimensions as well as another construction in dimension three which involves certain Calabi–Yau threefolds containing an Enriques surface. The constructions also show that potential density holds for (sufficiently) general members of the families.

We introduce a tensor decomposition of the $\ell$-adic Tate module of an abelian variety $A_0$ defined over a number field which is geometrically isotypic. If $A_0$ is potentially of $\GL_2$-type and defined over a totally real number field, we use this decomposition to describe its Sato--Tate group and to prove the Sato--Tate conjecture in certain cases., Comment: Minor changes. To appear in Math. Zeit

The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds for the translation length of a semisimple element x in SL_n(R) in terms of its associated Mahler measure. We use these geometric methods to prove the existence of extensions of number fields in which fixed sets of primes have certain prescribed splitting behavior.

In all available papers, on power integral bases of any pure sextic number fields K generated by a complex root α of a monic irreducible polynomial f ( x ) = x 6 − m ∈ Z [ x ] , it was assumed that the rational integer m ≠ ∓ 1 is square free. In this paper, we investigate the monogeneity of any pure sextic number field, where the condition m is a square free rational integer is omitted. We start by calculating an integral basis of Z K ; the ring of integers of K. In particular, we characterize when Z K = Z [ α ] , that is when Z K is monogenic and generated by α. We give sufficient conditions on m, which warranty that K is not monogenic. We finish the paper by investigating the case, where m = e 5 and e ≠ ∓ 1 is a square free rational integer.

Suppose an elliptic curve E / Q has good supersingular reduction at a prime p, and satisfies 1 + p − # E ( Z / p Z ) = 0 , which is always true if p is supersingular and p > 3 by the Hasse inequality. Expanding Kobayashi's plus/minus-local points over the cyclotomic Z p -extension of Q p , where π is any element in C p with positive p-adic valuation, we construct doubly indexed plus/minus local points over Q p ( π p n − m , μ p n ) (which we will call local Iwasawa Kummer extensions) as n , m vary. Then, we explore various possible plus/minus Selmer groups over Z p -extensions of number fields over which p is ramified, or doubly indexed plus/minus Selmer groups over Iwasawa Kummer extensions, and study or speculate their properties using the above plus/minus local points. Finally, again using the above points, we study the following: Suppose k is a ramified quadratic extension of Q p . We find a bound of [ E ( m k ( μ p n ) ) : E + ( m k ( μ p n ) ) + E − ( m k ( μ p n ) ) ] as n varies.

By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular., Comment: 14 pages

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

Let N N be one of the 38 38 distinct square-free integers such that the arithmetic group Γ 0 ( N ) + \Gamma _0(N)^+ has genus one. We constructed canonical generators x N x_N and y N y_N for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of x N x_N , which we express as a polynomial in y N y_N with coefficients that are rational functions in x N x_N . As a corollary, we prove that for any point e e in the upper half-plane which is fixed by an element of Γ 0 ( N ) + \Gamma _0(N)^+ , one can explicitly evaluate x N ( e ) x_N(e) and y N ( e ) y_N(e) . As it turns out, each value x N ( e ) x_N(e) and y N ( e ) y_N(e) is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of x N ( τ ) x_N(\tau ) and y N ( τ ) y_N(\tau ) at any CM point τ \tau , including elliptic points, for any genus one group Γ 0 ( N ) + \Gamma _0(N)^+ . Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.

Let $A$ be an abelian variety defined over a number field $k$, let $p$ be an odd prime number and let $F/k$ be a cyclic extension of $p$-power degree. Under not-too-stringent hypotheses we give an interpretation of the $p$-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of $A$ of the ${\rm Gal}(F/k)$-valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which the $p$-completion of the Mordell-Weil group of $A$ over $F$ is not a projective Galois module., Comment: Final version, 26 pages. To appear in Math. Proc. Camb. Phil. Soc

We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax-Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty-Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty-Kim theory to arbitrary number fields., 14 pages

We define a family of pseudodifferential operators on the Hilbert space $$L^2({\mathbb{Q}_p})$$ of complex valued square-integrable functions on the $$p$$ -adic number field $${\mathbb{Q}_p}$$ . These generalise the Vladimirov derivative, using the Dirichlet and other suitable multiplicative characters.The Riemann zeta-function and the related Dirichlet $$L$$ -functions can be expressed as a trace of these operators on a subspace of $$L^2({\mathbb{Q}_p})$$ . We also extend this to the $$L$$ -functions associated with modular (cusp) forms. Wavelets on $${\mathbb{Q}_p}$$ are common sets of eigenfunctions of all these operators.

Let H λ 4 be the Hecke group x , y : x 2 = y 4 = 1 and, for a square-free positive integer n , consider the subset ℚ ∗ − n = a + − n / c | a , b = a 2 + n / c ∈ ℤ , c ∈ 2 ℤ of the quadratic imaginary number field ℚ − n . Following a line of research in the relevant literature, we study the properties of the action of H λ 4 on ℚ ∗ − n . In particular, we calculate the number of orbits arising from this action for every such n . Some illustrative examples are also given.

A celebrated theorem of Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number $p$, we prove that there is an integral irreducible matrix with spectral radius $p$, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number $p$, there is an irreducible shift of finite type with entropy $\log(p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data., Comment: The referee's suggestions are incorporated; in particular an upper bound for the Perron--Frobenius degree is derived from the main result. See Theorem 1.6. To appear in Ergodic Theory and Dynamical Systems

Let q be a prime number and $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta $ a root of an irreducible trinomial $x^{6}+ax+b$ having integer coefficients. In this paper, we provide some explicit conditions on $a, b$ for which K is not monogenic. As an application, in a special case when $a =0$ , K is not monogenic if $b\equiv 7 \mod 8$ or $b\equiv 8 \mod 9$ . As an example, we also give a nonmonogenic class of number fields defined by irreducible sextic trinomials.

This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions., Comment: 21 pages. arXiv admin note: text overlap with arXiv:math/0408381 by other authors

A collection of varieties satisfies uniform potential density if each of them contains a dense subset of rational points after extending its ground field by a bounded degree. In this paper, we prove that uniform potential density holds for connected algebraic groups of a fixed dimension over fields of characteristic zero as well as elliptic K3 surfaces over number fields., Final version, accepted by International Mathematics Research Notices

Let O be the ring of integers of a number field, and let n ≥ 3 . This paper studies bi-interpretability of the ring of integers Z with the special linear group SL n ( O ) , the general linear group GL n ( O ) and the subgroup T n ( O ) of GL n ( O ) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

Class Field Theory (CFT) is the main achievement of algebraic number theory of the 20th century. Its reach, beauty and power, stemming from the first steps in algebraic number theory by Gauß, have substantially influenced number theory. Shafarevich wrote: 'Weil was undoubtedly right when he asserted, in the preface to the Russian edition of his book on number theory 1 , that since class field theory pertains to the foundation of mathematics, every mathematician should be as familiar with it as with Galois theory. Moreover, just like Galois theory before it, class field theory was reputed to be very complicated and accessible only to experts ... For class field theory, on the other hand, there is a wide range of essentially different expositions, so that it is not immediately obvious even whether the subject is the same'. 2 Weil's opinion has proved to be quixotic: these days even some number theorists are not familiar with the substance of CFT. This text reviews the enduring process of discovering new branches of CFT and its generalisations. Many of such developments were complicated at their early stages and some were difficult or impossible to understand for their contemporaries. Three main generalisations of CFT and their further extensions will be presented and some of their key fundamental features will be discussed. This text proposes eight fundamental problems. We start with Kummer theory, a purely algebraic exercise, whose highly non-trivial arithmetic analogues over arithmetic fields are supplied by CFT. Kummer theory is an algebraic predecessor of CFT including its existence theorem. Then we discuss the fundamental split of (one-dimensional) CFT into special CFT (SCFT) and general CFT (GCFT). This split has enormously affected many developments in number theory. Section 3 delves into four fundamental parts of CFT including the reciprocity map, existence theorem, explicit formulas for the Hilbert symbol and its generalisations, and interaction with ramification theory. Section 4 briefly touches on higher Kummer theory using Milnor K-groups, i.e. the norm residue isomorphism property. Three generalisations of CFT: Langlands correspondences (LC), higher CFT, and anabelian geometry are discussed in section 5. We note that the split of CFT into SCFT and GCFT is currently somehow reproduced at the level of generalisations of CFT: LC over number fields does not yet have any development parallel to GCFT, while higher CFT is parallel to GCFT and it does not have substantial developments similar to SCFT. In the last section we specialise to elliptic curves over global fields, as an illustration. There we consider two further developments: Mochizuki's inter-universal Teichmüller theory (IUT) which is pivoted on anabelian geometry and two-dimensional adelic analysis and geometry which uses structures of two-dimensional CFT. We also consider the fundamental role of zeta integrals which may unite different generalisations of CFT. Similarly to the situation with LC, the current studies of special values of zeta-and L-functions of elliptic curves over number fields, except two-dimensional adelic analysis and geometry, use special structures and are not of general type. There is no attempt to mention all the main results in CFT and all of its generalisations or all of their parts, and the text does not include all of bibliographical references. 1 [64] 2 in Foreword to [13].

This is a contribution toward calculating the K-theory of a certain class of affine algebraic surfaces which is made possible due to works by Cortinas, Haesemeyer, Schlichting, Walker and Weibel. We discover that the negative K-theory and higher reduced $${\widetilde{K}}_n$$ -group of the cone of a smooth projective plane curve over number field k, as vector spaces, are determined solely by the degree of the curve (and the ground field of course). In particular, the $${\widetilde{K}}_n$$ -groups of the cone for $$n\ge 2$$ alternate between zero and the Jacobian ring of the cone.

Let K be a number field, and let G be a finitely generated subgroup of $$K^\times $$ . We are interested in computing the degree of the cyclotomic-Kummer extension $$K(\root n \of {G})$$ over K, where $$\root n \of {G}$$ consists of all n-th roots of the elements of G. We develop the theory of entanglements introduced by Lenstra, and we apply it to compute the above degrees.

Kuroda’s formula relates the class number of a multiquadratic 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.

Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a finite extension $F$ of $K$ where $p$ is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell-Weil ranks of $E$ over a subextension of the cyclotomic $\mathbb{Z}_p$-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions., Comment: 20 pages

When $$A=3$$ , the positive integral solutions of the so-called Markoff equation $$\begin{aligned} M_A:x^2 + y^2 + z^2 = Axyz \end{aligned}$$ can be generated from the single solution (1, 1, 1) by the action of certain automorphisms of the hypersurface. Since Markoff’s proof of this fact, several authors have showed that the structure of $$M_A(R)$$ , when R is $${\mathbb Z}[i]$$ or certain orders in number fields, behave in a similar fashion. Moreover, for $$R={\mathbb Z}$$ and $$R={\mathbb Z}[i]$$ , Zagier and Silverman, respectively, have found asymptotic formulae for the number of integral points of bounded height. In this paper, we investigate these problems when R is a polynomial ring over a field K of odd characteristic. We characterize the set $$M_A(K[t])$$ in a similar fashion as Markoff and previous authors. We also give an asymptotic formula that is similar to Zagier’s and Silverman’s formula.

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups [3,2]. In this article, we prove similar result on isolating the cuspidal spectrum in [3] for the function field case., The paper has been accepted for publication in SCIENCE CHINA Mathematics. A gap in the previous version is fixed

Let $$K = \mathbb {Q} (\alpha )$$ be a pure number field generated by a complex root $$\alpha$$ of a monic irreducible polynomial $$F(x) = x^{20}-m$$ , with $$m \ne \mp 1$$ a square free rational integer. In this paper, we study the monogenity of K. We prove that if $$m\not \equiv 1\ \text{(mod } {4})$$ and $$\overline{m}\not \in \{\overline{1}, \overline{7}, \overline{18}, \overline{24}\} \ \text{(mod } {25})$$ , then K is monogenic. But if $$m\equiv 1\ \text{(mod } {16})$$ or $$m\equiv 1\ \text{(mod } {25})$$ , then K is not monogenic. Some illustrating examples are given too.

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [Formula: see text]. We prove that if [Formula: see text], then [Formula: see text] is monogenic. But if [Formula: see text] and [Formula: see text], then [Formula: see text] is not monogenic. Some illustrating examples are given.

Let $$K = \mathbb {Q} (\alpha )$$ be a pure number field generated by a root $$\alpha$$ of a monic irreducible polynomial $$F(x) = x^{3^r} -m$$ , with $$m \ne \pm 1$$ is a square-free rational integer and r is a positive integer. In this paper, we study the monogenity of K. We prove that if $$m \not \equiv \pm 1 \ \text { (mod }{9})$$ , then K is monogenic. We give also sufficient conditions on r and m for K to be not monogenic. Some illustrating examples are given too.

Let (X, L) be a polarized variety over a number field K. We suppose that L is an hermitian line bundle. Let M be a non compact Riemann Surface and $$U\subset M$$ be a relatively compact open set. Let $$\varphi :M\rightarrow X(\mathbf{C})$$ be a holomorphic map. For every positive real number T, let $$A_U(T)$$ be the cardinality of the set of $$z\in U$$ such that $$\varphi (z)\in X(K)$$ and $$h_L(\varphi (z))\le T$$ . After a revisitation of the proof of the sub exponential bound for $$A_U(T)$$ , obtained by Bombieri and Pila, we show that there are intervals of the reals such that for T in these intervals, $$A_U(T)$$ is upper bounded by a polynomial in T. We then introduce subsets of type S with respect of $$\varphi $$ . These are compact subsets of M for which an inequality similar to Liouville inequality on algebraic points holds. We show that, if M contains a subset of type S, then, for every value of T the number $$A_U(T)$$ is bounded by a polynomial in T. As a consequence, we show that if M is a smooth leaf of an algebraic foliation in curves defined over K then $$A_U(T)$$ is bounded by a polynomial in T. Let S(X) be the subset (full for the Lebesgue measure) of points which verify some kind of Liouville inequalities. In the second part we prove that $$\varphi ^{-1}(S(X))\ne \emptyset $$ if and only if $$\varphi ^{-1}(S(X))$$ is full for the Lebesgue measure on M.

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is [Formula: see text], but negative if the rank is [Formula: see text]. Moreover, for rank [Formula: see text] Drinfeld modules we classify those cases where the answer is positive.

Let $p\equiv -q \equiv 5\pmod 8$ be two prime integers. In this paper, we investigate the unit groups of the fields $ L_1 =\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q}, \sqrt{-1} )$ and $ L_1^+=\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q} )$. Furthermore , we give the second $ 2$-class groups of the subextensions of $L_1$ as well the $2$-class groups of the fields $ L_n =\mathbb{Q}( \sqrt{p}, \sqrt{q}, \zeta_{2^{n+2}} )$ and their maximal real subfelds., Comment: 18 pages

Algebraična teorija števil s konstrukcijo končne razredne grupe pove, v kakšni meri enolična faktorizacija v algebraičnih številskih poljih ne drži. Za nekatere izmed njih je značilna enolična dolžina faktorizacij posameznega elementa. Algebraic number theory introduces finite class group in order to determine the extent of factorization’s non-uniqueness in algebraic number fields. Some have a special property of unique factorization length for each element.