I definitely did not mean to suggest that ring theory or field theory not be developed at all! (Dispensing with algebra all together is clearly way too radical an approach that the mass public is not yet ready for, pedagogically.) I just don't think that it is reasonable to omit Galois theory from such a systematic course due to time concerns; at the very least it can be introduced and its applications alluded to. Also, +1 for "for self-education and single issue purposes you must do whatever it takes"

This of course works with polynomials with repeated roots, where you just throw 1/(x-r_i)^j for 1 <= j <= k with k the degree of repetition. Asking students to extend the case for Q with distinct roots to arbitrary Q might be a fun exercise.

@Wadim, I don't even know a rigorous definition of a game with an unboundable number of moves (a game with an infinite order of boundedness, as I've read about in J.H. Conway's note "More Infinite Games"), and that it might require logic does seem plausible.

@Wadim: My understanding of unboundedly unbounded games is that they cannot be formalized inductively as pairs of sets of games, a la Conway (hence they are not strictly combinatorial games). However, they (and their strategies) are still sensible though, such as the one in this question or, for a broader class of examples, back-and-forth (Ehrenfeucht-Fraisse) games on first-order equivalent structures.