How can I prove that $\pi+\pi^2$ is irrational?

Question

I am trying to prove that $\pi+\pi^2$ is irrational assuming that $\pi$ is transcendental.

My work: I noticed that at least one of $\pi+\pi^2$ and $\pi-\pi^2$ is transcendental. (Because algebraic numbers are closed under addition.) However, similar methods probably cannot resolve the question, because otherwise they could be used to show that $\pi+e$ is irrational which is an open problem. This makes me think that a solution must use the fact that the numbers $\pi$ and $\pi^2$ are related.

Not a solution of a polynomial equation with rational coefficients? (I know it was pretty stupid of me, but now I get it.) — Adam, May 16 '23 at 15:05

score 7 · Accepted Answer · answered May 16 '23 at 12:39

7

If $\pi+\pi^2$ were rational then $\pi$ would be algebraic.

answered May 16 '23 at 12:39

FShrike

40,125

How can I prove that $\pi+\pi^2$ is irrational?

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