Representation of numbers by quadratic forms in algebraic number fields

1988 ◽  
Vol 43 (5) ◽  
pp. 2663-2669
Author(s):  
E. A. Kashina
Keyword(s):  
Algebraic Number ◽  
Quadratic Forms ◽  
Number Fields ◽  
Algebraic Number Fields

On Lattices in a Module Over a Matrix Algebra

1966 ◽  
Vol 9 (1) ◽  
pp. 57-61 ◽  
Author(s):  
Nobuo Nobusawa
Keyword(s):  
Algebraic Number ◽  
Quadratic Forms ◽  
Matrix Algebra ◽  
Symmetric Matrix ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Left Module ◽  
Bilinear Mapping ◽  
The Matrix

Let A be the matrix algebra of type n × n over a finite algebraic number field F, and V the module of matrices of type n × m over F. V is naturally an A-left module. Given a non-singular symmetric matrix S of type m × m over F, we have a bilinear mapping f of V on A such that f(x, y) = xSy' for elements x and y in V where y' is the transpose of y. In this case, corresponding to the arithmetic of A([l]), the arithmetical theory of V will be discussed to some extent as we establish the arithmetic of quadratic forms over algebraic number fields ([2]). In this note, we shall define a lattice in V with respect to a maximal order in A. and determine its structure (Theorem 1), and after giving a structure of a complement of a lattice (Theorem 2), we shall give a finiteness theorem of class numbers of lattices under some assumption (Theorem 3).

scholarly journals Exponents of the class groups of imaginary Abelian number fields

1987 ◽  
Vol 35 (2) ◽  
pp. 231-246 ◽  
Author(s):  
A. G. Earnest
Keyword(s):  
Algebraic Number ◽  
Quadratic Forms ◽  
Class Group ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Abelian Field ◽  
The Ideal

It is a classical result, deriving from the Gaussian theory of genera of integral binary quadratic forms, that there exist only finitely many imaginary quadratic fields for which the ideal class group is a group of exponent two. This finiteness has been shown to extend to all those totally imaginary quadratic extensions of any fixed totally real algebraic number field. In this paper we put forward the conjecture that there exist only finitely many imaginary abelian algebraic number fields which have ideal class groups of exponent two, and we examine the extent to which existing methods can be brought to bear on this conjecture. One consequence of the validity of the conjecture would be a proof of the existence of finite abelian groups which do not occur as the ideal class group of any imaginary abelian field.

scholarly journals A Note on Units of Algebraic Number Fields

1955 ◽  
Vol 9 ◽  
pp. 115-118 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Existence Theorem ◽  
Algebraic Number ◽  
Present Note ◽  
Number Fields ◽  
Algebraic Number Fields

We shall prove in the present note a theorem on units of algebraic number fields, applying one of the strongest formulations, be Hasse [3], of Grunwald’s existence theorem.

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