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Let $t$ be transcendental over $\\G_p$, and let $K$ be a finite extension of $\\G_p(t)$. In this case $\\G_p[t]$ has a definition (with parameters) over $K$ of the form $\\forall \\exists \\ldots \\exists P$ with only one variable in the range of the universal quantifier and $P$ being a polynomial over $K$.\n  2. 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Among other results, we show that the following assertions are true.\n  1. Let $\\G_p$ be an algebraic extension of a field of $p$ elements and assume $\\G_p$ is not algebraically closed. Let $t$ be transcendental over $\\G_p$, and let $K$ be a finite extension of $\\G_p(t)$. In this case $\\G_p[t]$ has a definition (with parameters) over $K$ of the form $\\forall \\exists \\ldots \\exists P$ with only one variable in the range of the universal quantifier and $P$ being a polynomial over $K$.\n  2. 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