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 $\phi$ of the language of fields we construct a quantifier-free formula ${\phi }^{\mathrm{\prime }}$ such that $ACF⊢\phi \leftrightarrow {\phi }^{\mathrm{\prime }}$. We devote the last section of the paper to show some applications of this constructive version in mathematics and physics.