Abstract
Suppose I is a prime ideal in k[X1, ...,Xn] with a given finite generating set and k(q1,...,qm) is a finitely generated subfield of the field of fractions Z of k[X1, ..., Xn]/I and c is an element of Z. We present Groebner basis techniques to determine:
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if c is transcendental over k(q1,...,qm),
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a minimal polynomial for c if c is algebraic over k(q1,...,qm).
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the algebraic or transcendental nature of Z over k(q1,...,qm).
The information about c also tells whether c lies in k(q1,...,qm), solving the subfield membership problem. Determination of the algebraic or transcendental nature of Z over k(q1,...,qm) includes finding the index in case of algebraicity or transcendence degree in case the extension is transcendental. The determination of the nature of Z over k(q1,...,qm) is not simply an iterative application of the results for c and only requires computing one Groebner basis.
Supported in part by NSF & ARO through ACSyAM/MSI at Cornell University, #DAAL03-91-C-0027.
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Sweedler, M. (1993). Using Groebner bases to determine the algebraic and transcendental nature of field extensions: Return of the killer tag variables. In: Cohen, G., Mora, T., Moreno, O. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1993. Lecture Notes in Computer Science, vol 673. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56686-4_34
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DOI: https://doi.org/10.1007/3-540-56686-4_34
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