Excuse my typography - I'm new here... a, b, and c are rational numbers. I want to prove that * IF S = root(a) + root(b) + root(c) is rational THEN root(a), root(b) and root(c) are rational in themselves. Now I have done as follows: I reverse the problem and try to show that: * IF 0, 1 or 2 of root(a) etc are rational THEN the sum S must be irrational. If all three of the roots are rational then obviously the sum is too. This would conclude the proof. I can easily prove that if 1 or 2 of root(a) etc are rational, then the sum is, in fact, irrational. The trouble is that I can't prove the last case: * Show that IF none of root(a) root(b) and root(c) are rational, THEN their sum is irrational. I have tried the standard technique of assuming the sum is rational = m/n then rearranging terms and squaring successively to get rid of all root signs, then looking at parity but so far this seems to be a dead end - I can't draw ny conclusions from this. Could someone please help me with this - What insights do I need to attack this problem from some new angle? I need all help I can get here.