On completely multiplicative functions whose values are roots of unity

Let K be an absolutely abelian number field, and t be a positive integer relatively prime to $[K: \mathbb {Q}]$ . We provide a method to use the t -rank of the class group of K to obtain a lower bound on the t -rank of class groups of any cyclotomic fields (or real cyclotomic fields) that contain this K . As an application, for any odd integer $t> 1$ , we provide a family of cyclotomic fields whose class groups have t -rank at least $2$ . Furthermore, we prove that the class number of $\mathbb {Q}(\zeta _{p^e} + \zeta _{p^e}^{-1})$ cannot be equal to $2$ , where p is a prime, $e \geq 1$ is an integer, and $\zeta _{p^e}$ denotes a primitive $p^e$ -th root of unity. We also provide an upper bound for the class number of a real cyclotomic field.

Major progress in group theory occurred in the nineteenth century but the evolution of group theory began already in the latter part of 18th century. Some characteristic features of 19th century mathematics which had crucial impact on this evolution are concern for rigor and abstraction and the view that mathematics is a human activity without necessarily referring to physical situations. In 1770, Joseph Louis Lagrange (1736-1813) wrote his seminal memoir Reflections on the solution of algebraic equations. He considered ‘abstract’ questions like whether every equation has a root and, if so, how many were real/complex/positive/negative? The problem of algebraically solving 5th degree equations was a major preoccupation (right from the 17th century) to which Lagrange lent his major efforts in this paper. His beautiful idea (now going under the name of Lagrange’s resolvent) is to ‘reduce’ a general equation to auxiliary (resolvent) equations which have one degree less. Later, the theory of finite abelian groups evolved from Carl Friedrich Gauss’s famous “Disquisitiones Arithmeticae”. Gauss (1777-1855) established many of the important properties though he did not use the terminology of group theory. In his work, finite abelian groups4 Math Unlimitedappeared in different forms like the additive group Zn of integers modulo n, the multiplicative group Z∗n of integers modulo n relatively prime to n, the group of equivalence classes of binary quadratic forms, and the group of n-th roots of unity. In 1872, Felix Klein delivered a famous lecture A Comparative Review of Recent Researches in Geometry. The aim of his (so-called) Erlangen Program was the classification of geometry as the study of invariants under various groups of transformations. So, the groups appear here “geometrically” as groups of rigid motions, of similarities, or as the hyperbolic group etc. During the analysis of the connections between the different geometries, the focus was on the study of properties of figures invariant under transformations. Soon after, the focus shifted to a study of the transformations themselves. Thus the study of the geometric relations between figures got converted to the study of the associated transformations. In 1874, Sophus Lie introduced his “theory of continuous transformation groups” what we basically call Lie groups today. Poincaré and Klein began their work on the so-called automorphic functions and the groups associated with them around 1876. Automorphic functions are generalizations of the circular, hyperbolic, elliptic, and other functions of elementary analysis. They are functions of a complex variable z, analytic in some domain, and invariant under the group of transformations x′ = ax+b(a, b, c, d real or complex and ad − bc 0), or under some subgroup of this group. We end this introduction with the ancient problem of finding all those positive integers (called congruent numbers) which are areas of right-angled triangles with rational sides. To this date, the general result describing all of them precisely is not proved. However, the problem can be re-stated in terms of certain groups called elliptic curves and this helps in getting a hold on the problem and in obtaining several partial results. Indeed, if n is a positive integer, then the rational numbers x, y satisfying y2 = x3 − n2x form a group law which can be described geometrically. Then, n is a congruent number if and only if, this group is infinite!

Noether's problem for some semidirect products

A kind of fixed-point problem in the area of discrete tomography is proposed and investigated.Our chief concern in this paper is the case of square windows in the plane.Dealing with the arrays which are bounded, of polynomial growth, and finite-ring-valued, one comes across several interesting phenomena of combinatorial and arithmetic nature.

Power integral bases in prime-power cyclotomic fields

Introduction. It has long been surmised that every irreducible representation 2 of a group (M of order g can be written in the field of the g-th roots of unity. This question has attracted the efforts of a number of matheinaticians. Improving an earlier result of H. M11aschke [20],1 W. Burnside [14] proved that this conjecture is actually true for a very extensive class of representations. I. Schur [22] showed that it is correct for all soluble groups. The case of representations of odd degree with real character was treated by A. Speiser [25]12 The most important progress after that was made by H. Ilasse [11] who showed that the representation 2 can be written in the field of the g8-th roots of unity where n is a suitable positiverational integer depending on i. In the present paper, the old question is solved in its full generality for the first time. We prove the theorem:

Let $\ell$ be an odd regular prime, $k$ be the $\ell$th cyclotomic field, and $K=k(\sqrt[\ell]{a})$, where $a$ is a natural number that has exactly three distinct prime divisors. Assuming that there are exactly three places ramified in $K_\infty/k_\infty$, we study the $\ell$-component of the class group of the field $K$. For $\ell>3$, we prove that there always exists an unramified extension $\mathcal{N}/K$ such that $G(\mathcal{N}/K)\cong (\mathbb Z/\ell\mathbb Z)^3$, and all places over $\ell$ split completely in $\mathcal{N}/K$. If $\ell=3$ and $a$ is of the form $a=p^rq^s$, we give a complete description of the possible structure of the $\ell$-component of the class group of $K$. Some other results are also obtained.

Even-order subgroups of finite-dimensional division rings

We prove that for each multiplicative subgroup $A$ of finite index in $\mathbb{Q}^+$, the set of integers $a$ with $a, a+1 \in A$ is an IP-set. This generalizes a theorem of Hildebrand concerning completely multiplicative functions taking values in the $k$-th roots of unity.

Let R m be endowed with the Euclidean metric. The covering radius of a lattice Λ ⊂ R m is the least distance r such that, given any point of R m , the distance from that point to Λ is not more than r . Lattices can occur via the unit group of the ring of integers in an algebraic number field K , by applying a logarithmic embedding K ⁎ → R m . In this paper, we examine those lattices which arise from the cyclotomic number field Q ( ζ n ) , for a given positive integer n ≥ 5 such that n ≢ 2 ( mod 4 ) . We then provide improvements to a result of de Araujo in [3] , and conclude with an upper bound on the covering radius for this lattice in terms of n and the number of its distinct prime factors. In particular, we improve [3, Lemma 2] , and show that, asymptotically, it can be improved no further.

In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers N as limiting values of q-series as q → ζ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of N by analogous structures in the integer partitions P. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new q-series density formulas for any subset of N. To do so, we outline a theory of q-series density calculations from first principles, based on a statistic we call the "q-density" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.

Let p be a prime with p≡3 (mod 4), let n be an odd natural number and let . Consider the crosscorrelation funtion C d (t)=∑ i =1 pn−1 ζ ai − adi−t where ζ≠1 is a complex p-th root of unity and (a i ) is a maximal linear shift register sequence. In 7 the bound has been computed for p = 3. In this note we generalize this to for p≥ 3. Furthermore we giv an upper bound for the probability of the crosscorrelation function achieving the maximum absolute value.

For a finite system $$A = {\left\{ {a_{s} + n_{s} \mathbb{Z}} \right\}}^{k}_{{s = 1}}$$ of arithmetic sequences the covering function is w(x) = |{1 ≤ s ≤ k : x ≡ as (mod ns)}|. Using equalities involving roots of unity we characterize those systems with a fixed covering function w(x). From the characterization we reveal some connections between a period n0 of w(x) and the moduli n1, . . . , nk in such a system A. Here are three central results: (a) For each r=0,1, . . .,nk/(n0,nk)−1 there exists a J c {1, . . . , k−1} such that $${\sum\nolimits_{s \in J} {1/n_{s} = r/n_{k} } }$$. (b) If n1 ≤···≤nk−l <nk−l+1 =···=nk (0 < l < k), then for any positive integer r < nk/nk−l with r ≢ 0 (mod nk/(n0,nk)), the binomial coefficient $${\left( {\begin{array}{*{20}c} {l} \\ {r} \\ \end{array} } \right)}$$ can be written as the sum of some (not necessarily distinct) prime divisors of nk. (c) max(x∈ℤw(x) can be written in the form $${\sum\nolimits_{{\left( {s = 1} \right)}}^k {m_{s} /n_{s} } }$$ where m1, . . .,mk are positive integers.

Catalan’s Conjecture predicts that 8 and 9 are the only consecutive perfect powers among positive integers. The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu. A deep theorem about cyclotomic fields plays a crucial role in his proof. Like Fermat’s problem, this problem has a rich history with some surprising turns. The present article surveys the main lines of this history and outlines Mihăilescu’s brilliant proof.

Let k k be any imaginary abelian field, R R the integral group ring of G = Gal ( k / Q ) G = {\text {Gal}}(k/\mathbb {Q}) , and S S the Stickelberger ideal of k k . Roughly speaking, the relative class number h − {h^ - } of k k is expressed as the index of S S in a certain ideal A A of R R described by means of G G and the complex conjugation of k ; c − h − = [ A : S ] k;{c^ - }{h^ - } = [A:S] , with a rational number c − {c^ - } in 1 2 N = { n / 2 ; n ∈ N } \frac {1} {2}\mathbb {N} = \{ n/2;n \in \mathbb {N}\} , which can be described without h − {h^ - } and is of lower than h − {h^ - } if the conductor of k k is sufficiently large (cf. [6, 9, 10]; see also [5]). We shall prove that 2 c − 2{c^ - } , a natural number, divides 2 ( [ k : Q ] / 2 ) [ k : Q ] / 2 2{([k:\mathbb {Q}]/2)^{[k:\mathbb {Q}]/2}} . In particular, if k k varies through a sequence of imaginary abelian fields of degrees bounded, then c − {c^ - } takes only a finite number of values. On the other hand, it will be shown that c − {c^ - } can take any value in 1 2 N \frac {1} {2}\mathbb {N} when k k ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.