# two real closed fields- algebraic elements

If R_1\subset R_2 are two real closed fields (R_2 is an extension of R_1), then is it always the case that R_1 contains {R_2}_alg; By the latter I mean algebraic elements of R_2.

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What do you mean by "algebraic elements of $R_2$"? Do you mean those elements that are algebraic over $R_1$? Then the answer is yes. There is a straightforward proof: the only algebraic extension of $R_1$ is $R_1[i]$, and $i \notin R_2$. (where $i^2 = -1$)
As far as I know, "algebraic" without an "over $k$" clause, always means algebraic over the prime field, in this case $\mathbb Q$. – Andreas Blass Dec 12 '11 at 19:25