Open Access
Infinite objects in constructive mathematics
Thierry Coquand
- 01 Jan 2005
TL;DR: In this paper, the Axiom of Choice Theorem is used to prove the existence of an object satisfying a simple concrete property, but it is not clear if this proof gives a way to compute this object.
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Abstract: commutative algebra has become one of the “less computational” part of mathematics (if one looks at the proofs). One of the rare part which uses the general form of the Axiom of Choice Theorem: (Krull) An element is nilpotent iff it belongs to all prime ideals Theorem: Any field has an algebraic closure, unique up to isomorphism If we prove in commutative algebra the existence of an object satisfying a simple “concrete” property, it is not clear if this proof gives a way to compute this object
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Dynamical method in algebra: Effective Nullstellens\"atze
TL;DR: This work gives a general method for producing various effective Null and Positivstellensatze and constructive versions of abstract classical results of algebra based on Zorn's lemma in algebraically closed valued fields and ordered groups.
87
Generating non-Noetherian modules constructively
TL;DR: In this article, Heitmann gave a proof of a Basic Element Theorem, which has as corollaries some versions of the "Splitting-off" theorem of Serre and the Forster-Swan theorem in a non Noetherian setting.
54
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Essays in Constructive Mathematics
Harold M. Edwards
- 30 Nov 2004
TL;DR: In this paper, a fundamental theorem of algebraic geometry, the minimal splitting of polynomials with integer coefficients, is presented. But the proof of the fundamental theorem is not yet complete.
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