A constructive proof of Tarski's theorem on quantifier elimination in the theory of ACF

Assume that $ACF$ denotes the theory of algebraically closed fields. The renowned theorem of A. Tarski states that $ACF$ admits quantifier elimination. In this paper we give a constructive proof of Tarski's theorem on quantifier elimination in $ACF$. This means that for a given formula $φ$ of the language of fields we construct a quantifier-free formula $φ'$ such that $ACF\vdashφ\leftrightarrowφ'$. We devote the last section of the paper to show some applications of this constructive version in mathematics and physics.

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