On class groups of abelian extensions

Until recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.In this paper, we study in particular the cyclotomic fields of composite conductor.

We prove that if a complex valued completely multiplicative function F and a positive integer l≦5 satisfy the condition F(N) = Ul, where Ul is the set of all l-th roots of unity, then {F(n+1) F(n) ∣ ne N} = Ul.

Noether's problem for some semidirect products

We consider the problem of determining all power integral bases for the maximal real subfieldQ(ζ + ζ−1) of thep-th cyclotomic fieldQ(ζ), wherep≥ 5 is prime and ζ is a primitivep-th root of unity. The ring of integers isZ[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove thatifor one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators forp≤ 11, and it is plausible that there are no additional generators forp> 13 as well. Forp= 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 <p< 1000.

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!

In this chapter we shall investigate several types of abelian extension fields. First, we shall consider cyclotomic fields over the field of rational numbers and we shall determine their dimensionalities and Galois groups. Next we shall consider Kummer extensions, which are obtained by adjoining the roots of a finite number of pure equations x m = α to a field containing m distinct m-th roots of 1. Finally, we shall study the so-called abelian p-extensions, which are defined to be abelian extensions of p f dimensions of a field of characteristic p ≠ 0. The theory of characters of finite commutative groups is a basic tool for the investigation of Kummer extensions and abelian p-extensions. Besides this, our study of abelian p-extensions will be based on a certain type of ring, a ring of Witt vectors which can be constructed from any commutative algebra U over a field of characteristic p ≠ 0. For any such U and integer m = 1, 2, ···, we have a ring of Witt vectors M m(U) of characteristic p m . In the theory of valuations it is useful to pass to the limit as m → ∞ and to consider also rings M(U) of infinite Witt vectors. This will be considered in Chapter V.

Let $N/\mathbb{Q}$ be a real cyclic and tame extension of prime degree $l$ with $\Gamma=\mathscr{G}al(N/\mathbb{Q})$. We give the Hom description of the class of the torsion-free part of the group of units in $N$ in the class group of the order $\mathbb{Z}\Gamma/ (\sum_{\gamma \in \Gamma}\gamma )$. This representation depends only on the structure of the ideal class group of $N$ and determines the Galois module structure of the torsion-free part of the group of units in $N$ as an ideal of the $l$th cyclotomic field. Using this approach we derive necessary and sufficient conditions for all real and tame cyclic fields of prime degree to have Minkowski units. We extend also the class of known cyclic real fields with Minkowski units.

In this paper, for a totally real number field k we show the ideal class group of k(?n>07n)+ is trivial. We also study the p-component of the ideal class group of the cyclotomic Zp-extension.

Power integral bases in prime-power cyclotomic fields

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.

Triple symbols are arithmetic analogues of the mod n triple linking number in topology, where $$n &gt; 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is an integer. In this paper, we introduce a cohomological formulation of a mod n triple symbol for characters over a number field containing a primitive n -th root of unity. Our definition is motivated by the arithmetic Chern–Simons theory and in this respect it differs from earlier approaches to triple symbols. We show that our symbol agrees with that of Rédei when $$n=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and of Amano–Mizusawa–Morishita when $$n=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> .

Let F F be a finite field, A = F [ T ] A=F[T] , and k = F ( T ) k=F(T) . Let K m = k ( Λ m ) K_{m}=k(\Lambda _{m}) be the field extension of k k obtained by adjoining the m m -torsion on the Carlitz module. The class number h m h_{m} of K m K_{m} can be written as a product h m = h m + h m − h_{m}=h_{m}^{+}h_{m}^{-} . The number h m − h_{m}^{-} is called the relative class number. In this paper a formula for h m − h_{m}^{-} is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of p p -th roots of unity. Some consequences of this formula are also derived.

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.

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 $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\mbox{Prym}(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\mbox{Prym}(\widetilde{C}_n/C_n)$ as a Prym variety and determine the interesting simple factors other than elliptic curves or hyperelliptic curves with small genus which come up in $J_n$ so that the endomorphism rings contains the totally real field $\mathbb{Q}(\zeta_n+\zeta^{-1}_n)$.