CICM 2010: Intelligent Computer Mathematics pp 189-203

# A Formal Quantifier Elimination for Algebraically Closed Fields

• Cyril Cohen
• Assia Mahboubi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6167)

## Abstract

We prove formally that the first order theory of algebraically closed fields enjoys quantifier elimination, and hence is decidable. This proof is organized in two modular parts. We first reify the first order theory of rings and prove that quantifier elimination leads to decidability. Then we implement an algorithm which constructs a quantifier free formula from any first order formula in the theory of ring. If the underlying ring is in fact an algebraically closed field, we prove that the two formulas have the same semantic. The algorithm producing the quantifier free formula is programmed in continuation passing style, which leads to both a concise program and an elegant proof of semantic correctness.

## Keywords

Decision Procedure Order Theory Great Common Divisor Formal Polynomial Disjunctive Normal Form
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