Showing papers on "Ring of integers published in 1983"

Unramified class field theory of arithmetical surfaces

Abstract: Let k be an algebraic number field, (9k its ring of integers and V a non-empty open subscheme of Spec(Ck). Let X be a projective smooth geometrically irreducible scheme over k, and X a regular proper flat scheme over V such that X x Vk X. The purpose of this paper is to give an explicit description of the abelian fundamental groups ,ab (X) and 7T b(X), using some "idele class groups" attached to X and X by a K-theoretical method. Its outline is as follows: In general, for a noetherian scheme Z and i = 0 or 1, we define the group SKi(Z) to be the cokernel of

Totally positive units and squares

Abstract: Let K be a finite cvclic extension of the rational number field Q. with Galois group G( K/Q) of order p" for an odd prime p. Armitage and Frohlich [1] proved that if the order of 2 modulo p is even and the class number hA of K is odd then L"A = U-. where (," is the group of units of the ring of integers CA of K. UsL is the group of totally positive units, and ' is the group of unit squares. The purposc of this paper is to provide a generalization of this result to a larger class of ahelian extensions of Q.2 1. We begin with some definitions and notation. Let K be a finite real abelian extension of Q. We define UI0 to be {u E UK-: u k_ mod(4) for some k E CA:), and set U= Uk," where UK is the group of units of the ring of integers CE of K and U,is the group of unit squares. Similarly we set U = U(4 /U-, where UK is the group of totally positive units. It is worth noting at this juncture that the units 11 in UA) are precisely those units for which K(v'i)IK is ramified at, at most, the infinite K-primes when I K: QI is odd. This fact follows from Hecke [5] and Kummer theory considerations. We let FK denote the group of cyclotomic units a la Leopoldt [8]. We caution the reader that these are not Hasse's circular units, CK (see [4]). FK is a subgroup of UA related to CK, and Leopoldt has obtained the result I UK: FK I= h KQ(; where Q,' is an integer depending on the structure of G(K/Q) and hA is the class number of K. For the reader who is interested in an easily understood exposition of Leopoldt's work in this direction we suggest Oriat's description [10] as an alternative to [8]. Now, FZ, F'? FA and F2 are defined in an analogous fashion to that of UA. We let K ") denote the Hilbert class field of K; i.e., I KV): K l hK, and we let K") denote the "narrou'v class field of K, i.e., G(K(+'/K) is the quotient group of ideals of CA modulo totally positive principal ideals. We note that asking when U( = UAis equivalent to asking when K(t) = K(. This fact, for real K, is the statement of [7, Theorem 3.1, p. 203], the proof of which uses the Artin map. 2. To prove the main result we first need two lemmas. The first lemma is provided in its most general form since it may be of independent interest. In the following lemma i's, denotes the field of 2 elements. Received by the editors October 15 1981 and, in revised form. JulyV 13, 1982 198() MAatlienatt(s Suhlect (Cus.ifiation. Primarv I2A65. 12A95. Kei uortAsajnlphuase. Totallv positive units. squares. cvclotornic units, class field theorv. 'This author's research is supported by N.S.E.R.C. Canada University rcsearch fellowship = U0077 2Kummer began the investigation of U` and U' for K Q(Fp + F I) where Fp denotes a primitive ) ti root of unity The classification of those p for which L/UA =/ remains unsolved Our main result advances the solution of this problem as well 1983 Aimerican Mathematical Societv (XX)2-9939X'82 /(X)(X)-()736 /$02 (H)

Parallel Computation for Well-Endowed Rings

Abstract: It is shown that a probabilistic Turing acceptor or transducer running within space bound S can be simulated by a time S 2 parallel machine and therefore by a space S 2 deterministic machine. (Previous simulations ran in space $6.) In order to achieve these simulations, known algorithms are extended for the computation of determinants in small arithmetic parallel time to computations having small Boolean parallel time, and this development is applied to computing the completion of stochastic matrices. The method introduces a generalization of the ring of integers, called well-endowed rings. Such rings possess a very efficient parallel implementation of the basic (+, -, ×) ring operations.

Essentially closed radical classes

Abstract: The main goal of this paper is to describe radical classes closed under essential extensions. It turns out that such classes are precisely the homomorphically closed semisimple classes, and hence a radical class is essentially closed if and only if it is subdirectly closed. Moreover, a class is closed under homomorphic images, direct sums and essential extensions if and only if it is an essentially closed radical class. Also radical classes are investigated which are closed under Dorroh essentially extensions only, such a radical class R consists of idempotent rings provided that R does not contain the ring of integers, meanwhile all the other radicals satisfy this requirement. A description of (hereditary and) Dorroh essentially closed radicals is given in Theorem 4.

On the class number of a unit lattice over a ring of real quadratic integers

Abstract: Let K be a totally real algebraic number field. In a positive definite quadratic space over K a lattice En is called a unit lattice of rank n if En has an orthonormal basis {e 1 …, en }. The class number one problem is to find n and K for which the class number of En is one. Dzewas ([1]), Nebelung ([3]), Pfeuffer ([6], [7]) and Peters ([5]) have settled this problem.

Relative Galois module structure of rings of integers and elliptic functions

Abstract: Let K be a quadratic imaginary number field with discriminant less than −4. For N either a number field or a finite extension of the p -adic field p , we let N denote the ring of integers of N . Moreover, if N is a number field then we write for the integral closure of [½] in N . For an integral ideal & of K we denote the ray classfield of K with conductor & by K (&). Once and for all we fix a choice of embedding of K into the complex numbers .

GROUP SCHEMES OF PERIOD p

Abstract: This paper gives an explicit construction of the finite commutative group schemes of period p defined over the ring of integers of the algebraic closure of the field Qp, and describes them in terms of a category of modules. Bibliography: 4 titles.

110. A Note on P.Rings

Abstract: Preliminary definitions. If a is an element of the F-ring M, then (a) denotes the principal ideal generated by a. If S is a subset of M, we call S an sp-system if S=i or a eS implies (a)S=. A nonempty subset S ofM is called a g-sp-system if S contains an sp-system .S’ such that g(x) S’=O for every element x of S, where S’ is called a kernel of S. An ideal I ofM is said to be g-halfprime if C(I)=M\I is a g-sp-system. Example. Consider Z, the ring of integers, as a F-ring with F =Z. Let p, q be two distinct prime numbers. Define g(a)-({a, pq}). Now g(pq)=(pq) and hence C((pq)) is g-sp-system with kernel C((pq)), which is not a g-system.