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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.

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# Proof that sum of 3 roots of rationals is rational etc

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