Gröbner Bases Applied to Finitely Generated Field Extensions

  title={Gr{\"o}bner Bases Applied to Finitely Generated Field Extensions},
  author={J{\"o}rn M{\"u}ller-Quade and Rainer Steinwandt},
  journal={J. Symb. Comput.},
Using a constructive field-ideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensionsk (x)/k(g), resp. how to determine the (separable) degree if k(x)/k(g) is algebraic. Moreover, this correspondence is used to derive a method for computing minimal polynomials and deciding field membership. Finally, a connection between certain intermediate fields of k(x)/k(g) and a minimal primary decomposition of a… 
On Dru\.zkowski's morphisms of cubic linear type
We use theorems of Muller-Quade and Steinwandt, Scheja and Storch, and van der Waerden to study Druzkowski's morphisms of cubic linear type with invertible Jacobian. In particular, we compare the
Realization Theory for Rational Systems: Minimal Rational Realizations
The study of realizations of response maps is a topic of control and system theory. Realization theory is used in system identification and control synthesis.A minimal rational realization of a given
Algorithms for fields and an application to a problem in computer vision
An investigation of an important problem in computer vision and the examination of the natural action of the group PGL_{m+1} x S_n on the set of n-point configurations leads to an investigation of algorithms for fields and an algorithm for testing whether a field is algebraically closed in another field.
Implementation techniques for fast polynomial arithmetic in a high-level programming environment
This paper discusses how to achieve high performance in implementing some well-studied fast algorithms for polynomial arithmetic in two high-level programming environments, AXIOM and Aldor.
Symmetric functions and the phase problem in crystallography
The calculation of crystal structure from X-ray diffraction data requires that the phases of the structure factors (Fourier coefficients) determined by scattering be deduced from the absolute values
Realization theory for rational systems
In this paper we solve the problem of realization of response maps for rational systems. Sucient and necessary conditions for a response map to be realizable by a rational system are presented. The
On computing a separating transcendence basis
A method is given to compute the transcendence degree and a separating transcendence basis of this field extension by means of simple linear algebra techniques.
Modelling, Analysis and Simulation Realization theory for rational systems
In this paper we solve the problem of realization of response maps for rational systems. Sufficient and necessary conditions for a response map to be realizable by a rational system are presented.


Using Groebner Bases to Determine the Algebraic and Transcendental Nature of Field Extensions: Return of the Killer Tag Variables
This work presents Groebner basis techniques to determine the algebraic or transcendental nature of Z over k(q1,...,qm), and a minimal polynomial for c if c is algebraic over k (q1), which is a finitely generated subfield of the field of fractions Z.
Basic Algorithms for Rational Function Fields
By means of Grobner basis techniques algorithms for solving various problems concerning subfields K(g) of a rational function field K(x):=K(x1, ?,xn) are derived, the essential idea is to reduce these problems to questions concerning an ideal of a polynomial ring.
Gröbner Bases and Primary Decomposition of Polynomial Ideals
An Algorithm to Determine Properties of Field Extensions Lying over a Ground Field
Let K L N be elds such that N is nitely generated over K. We will use Grr obner basis methods to calculate the transcendence degree of NjL and the degree N:L] if the extension is algebraic. There is
Deciding linear disjointness of finitely generated fields
Methods for e ectively deciding linear disjointness and freeness for elds lying under a nitely generated eld k(X) = Quot(k[X1; : : : ; Xs]=I(X)).
Computing Dimension and Independent Sets for Polynomial Ideals
Converting Bases with the Gröbner Walk
An algorithm which converts a given Grobner basis of a polynomial ideal I to a Grobners basis of I with respect to another term order is presented.
An implicitization algorithm with fewer variables
Gröbner bases - a computational approach to commutative algebra
This chapter discusses linear algebra in Residue Class Rings in Vector Spaces and Modules, and first applications of Gr bner Bases.
Foundations of Algebraic Geometry
Algebraic preliminaries Algebraic theory of specializations Analytic theory of specializations The geometric language Intersection-multiplicities (special case) General intersection-theory The