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Homework Statement
A non-transcendental number is one that's a root of a (non-constant) polynomial with rational coefficients.
Does allowing radicals as coefficients, eg: 5√3, 2^(1/3) get us any new different numbers?
Homework Equations
The Attempt at a Solution
1. In some cases we get no new numbers, eg:
[itex]x^2+\sqrt[3]{3}=0\Leftrightarrow x^6=3[/itex]
[itex]x^2+x\sqrt{3}+\sqrt{2}=0\Leftrightarrow x^2+x\sqrt{3}=-\sqrt{2} \Rightarrow x^4+3x^2+2x^3\sqrt{3}=2 \Leftrightarrow x^4+3x^2-2=-2x^3\sqrt{3} \Rightarrow (x^4+3x^2-2)^2=4x^6 \times 3[/itex]
We solve this by isolating the radicalss and squaring/cubing/etc them. But with more coefficients it becomes harder. What about a 5th degree polynomials with only cubic radical coefficients?
So the question is:
-Do we get new non-trascendental numbers if we allow rational AND radicals as coefficients.
-Can we always turn a polynomial with radical coefficients into a one with rational coefficients?
PS: By radicals I mean just some number that's written with the nth-root symbol. Use whatever definition you feel fits best.
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