Can the axioms for real closed fields be weakened in this way?

Question

I am told that real closed fields $(F,+,-,*,0,1,<)$ can be axiomatized by the axioms for ordered fields and also an axiom stating that every positive element has a square root and an axiom schema stating that every odd-degree polynomial has a root. I am wondering whether this can be weakened somewhat. Would it suffice to say, instead of the axiom schema for odd-degree polynomials, that for every odd natural number $n$, every element of $F$ has an $n$-th root? And if not, can someone exhibit a model of this weaker theory that is not a real-closed field?

Answer 1

Let $R$ be the ordered subfield of $\mathbb{R}$ consisting of all real numbers which are constructible by radicals over $\mathbb{Q}$ (the union of all radical extensions of $\mathbb{Q}$ contained in $\mathbb{R}$). This is an ordered field in which every positive element has a square root and every element has an $n$th root for all odd $n$. But it is not real closed since (by Abel-Ruffini / Galois) there are polynomials of degree $5$ over $\mathbb{Q}$ with real roots which are not constructible by radicals.