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Re: Simple question about irrationals, with a short note in the margin. :)
Posted:
Jan 14, 2014 9:54 PM


On Wed, 15 Jan 2014, Port563 wrote: > >> Can an algebraic irrational raised to a transcendental be an algebraic > >> irrational? > >> > >> Can an algebraic irrational raised to a transcendental be rational? > >> > >> Can a transcendental raised to a transcendental be an algebraic > >> irrational? > >> > >> Can a transcendental raised to a transcendental be rational? > > Which, sadly, was the skeleton upon which I built the original post. > > I am unaware of proofs or examples to settle the above, but am also aware > that I am decades out of date. > > I am relatively sure (in intuitive terms, that is) what are the answers: > > Can an algebraic irrational raised to a transcendental be an algebraic > irrational? YES, FAIRLY SURE > > Can an algebraic irrational raised to a transcendental be rational? YES, > FAIRLY SURE > > Can a transcendental raised to a transcendental be an algebraic irrational? > YES, VERY SURE > > Can a transcendental raised to a transcendental be rational? YES, VERY SURE > > Any comments if your intuition indicates similarly or otherwise?
I'd conjecture they're are all transcendental. Is it an open problem?



