For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998; A weak countable choice principle; available from [1]Archived 2020-02-19 at the Wayback Machine.
See Fred Richman; 1998; The fundamental theorem of algebra: a constructive development without choice; available from [2]Archived 2020-02-19 at the Wayback Machine.
Smale writes, "...I wish to point out what an immense gap Gauss's proof contained. It is a subtle point even today that a real algebraic plane curve cannot enter a disk without leaving. In fact, even though Gauss redid this proof 50 years later, the gap remained. It was not until 1920 that Gauss's proof was completed. In the reference Gauss, A. Ostrowski has a paper which does this and gives an excellent discussion of the problem as well..."
For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998; A weak countable choice principle; available from [1]Archived 2020-02-19 at the Wayback Machine.
See Fred Richman; 1998; The fundamental theorem of algebra: a constructive development without choice; available from [2]Archived 2020-02-19 at the Wayback Machine.
