A Formal Quantifier Elimination for Algebraically Closed Fields

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


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.


Decision Procedure Order Theory Great Common Divisor Formal Polynomial Disjunctive Normal Form 
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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Cyril Cohen
    • 1
  • Assia Mahboubi
    • 1
  1. 1.INRIA Saclay – Île-de-France, LIX École Polytechnique, INRIA Microsoft Research Joint Centre 

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