The following theorem is proved: Let R be an algebra with involution over an uncountable field F. Then if the symmetric elements of R are algebraic, R is algebraic. In this paper we consider the following question: "Let R be an algebra with involution over a field F, and assume that the symmetric elements S of R are algebraic over F. Is R algebraic over F?" Previous results related to this question have been obtained by restricting the kind of algebraic relationships satisfied by the symmetric elements. For example, it was shown by Baxter and Martindale [1] for fields of characteristic not 2, and later by the author [5] for arbitrary fields, that if the symmetric elements are algebraic of bounded degree (or more generally, satisfy a polynomial identity), then R must be algebraic. Another such result concerns rings whose symmetric elements are periodic (that is, for each seS, there is some integer n(s)> 1 such that sn(s)=s). In this case, the author has shown [6], [7] that R must be algebraic; in fact R satisfies a polynomial identity. When R is a division ring, much more can be said: I. N. Herstein and the author [2] have shown that R must actually be commutative. Finally, it has been shown by Osborn [8] that if S is nil and F is uncountable, then R is nil. This answers for uncountable fields a question of McCrimmon [4, p. 391]: "If S is nil, is R nil?" An affirmative answer to this question in general would follow from an affirmative answer to the first question. For, as has been observed by both McCrimmon [4, p. 390] and Osborn [8, p. 306], if S is nil then R must be a radical ring. But if R is algebraic, every element of the radical is nil; thus R would be nil. The result presented here differs from those described above in that no additional restrictions are imposed on the symmetric elements. We prove: THEOREM. Let R be an algebra with involution over an uncountable field F. Then if the symmetric elements of R are algebraic, R is algebraic. Received by the editors January 8, 1971. AMS 1969 subject classifications. Primary 1658.