Hallo, in the PS to his mail on duality, John Baez wrote:
Briefly, while the existence of an algebraic closure of Q can be shown without choice, it uniqueness-up-to-isomorphism seems to require choice. Also, while arithmetic operations in Qbar are computable, they seem to present interesting challenges.
it would look interesting to me to show that uniqueness up to isomorphism requires some kind of choice, e.g. by proving that it implies some choice principle. I started thinking about this and came to the following observation: Let K be the field obtained from Q by adjoining all square roots of (positive) primes or - equivalently - of all positive rational numbers. Then an algebraic closureof K is obviously the same as an algebraic closure of Q - even without choice. Now consider an arbitrary sequence of two element sets S(n), w.l.o.g. pairwise disjoint, and let S be their union. Now consider the polynomial ring Q(S) over Q in variables x(s) for all s in S. For each natural number n let p(n) be the n-th prime and consider the poynomials f_n:=x(s)+x(t) and g_n:=x(s)x(t)+p(n) in Q(S) where S(n) consists of the two elements s and t. Now let K' be the factor ring obtained from Q(S) by dividing out all f_n and g_n. If we have a choice function which assigns to each n an element s(n) in S(N), then there is a unique isomorphism from K' to K that maps each s(n) to the (positive) square root of p(n). Conversely, if we have an arbitrary isomorphism j from K' to K, we get a choice function which chooses for the each n unique s(n) in S(n) with j(n)>0. Thus existence of an isomorphism is equivalent to existence of a choice function. If the existence of an algebraic closure of K' could be shown without choice, then an isomorphism fom this algebraic closure to the set of algebraic complex numbers would restrict to an isomorphism between K and K' and thus render a chioce function for the S(n). Thus the uniqueness-up-to-isomorphism would imply choice for countable families of two-element sets. Maybe refinement oft his argument could even be used to get stronger choice principles. Greetings Reinhard Boerger