Does anyone know an example of a transcendental number raised to another transcendental number being algebraic? I'm tempted to go to Euler's formula (e^(πi) = -1) but can't prove that e^i is transcendental (I know Wolfram alpha says it is). Anyone have a simpler example?
Also, if you have some insight to this question, it would be much appreciated:
if a and b are algebraic numbers over a field, find the polynomial for which a+b and the polynomial for which a x b are solutions (zeros). I.e, if a satisfies p(x) = sum(i = 0, n) (a_i x^i) = 0 and b satisfies g(x) = sum(i = 0, n) (b_i x^i) = 0, find the polynomials h(x) and f(x) for which h(a +b) = 0 and f(a x b) = 0. I know there has to be a theorem out there about this, but I can't find it (but please don't use Galois Theory because I am not there yet).
Also, if you have some insight to this question, it would be much appreciated:
if a and b are algebraic numbers over a field, find the polynomial for which a+b and the polynomial for which a x b are solutions (zeros). I.e, if a satisfies p(x) = sum(i = 0, n) (a_i x^i) = 0 and b satisfies g(x) = sum(i = 0, n) (b_i x^i) = 0, find the polynomials h(x) and f(x) for which h(a +b) = 0 and f(a x b) = 0. I know there has to be a theorem out there about this, but I can't find it (but please don't use Galois Theory because I am not there yet).
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I'd stick with e^(πi) = -1.
Don't make this too tough; all you need is e and πi both being transcendental.
(That πi is transcendental is a quick proof by contradiction:
If πi were algebraic, then πi/i = π is also algebraic, which is a contradiction.)
I hope this helps!
Don't make this too tough; all you need is e and πi both being transcendental.
(That πi is transcendental is a quick proof by contradiction:
If πi were algebraic, then πi/i = π is also algebraic, which is a contradiction.)
I hope this helps!