I want to prove that all numbers of the form $a\sqrt{2}+b\sqrt{3}$ are irrational, where $a,b$ are rational (and not both zero). I'm not sure how to go about it. Find minimal polynomial, and show it is irreducible? I'm fairly new to abstract algebra. Is this the right way to go about it? Is there an easier way?
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Assume it’s rational and square it. What do you now know? – Robert Shore Jul 21 '22 at 22:31
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4Does this answer your question? Prove that for rational nonzero $a, b,$ the expression $a\sqrt3 + b\sqrt5$ is irrational. - found using an Approach0 search. It uses $\sqrt{5}$ instead of $\sqrt{2}$, but the procedure is the same. Note it's a bit simpler to instead use $a\sqrt{2} = r - b\sqrt{3}$, where $r$ is assumed rational, then square both sides to get a contradiction. – John Omielan Jul 21 '22 at 22:31
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1Yes, @JohnOmielan and Robert, this answers my question. Thank you. – Doug Jul 21 '22 at 23:05