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I know about algebraic numbers and transcendental numbers. How the roots of a polynomial with irrational coefficients are classified. Are they transcendental?

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    They are not necessarily transcendental: Take any polynomial with rational coefficients and multiply its coefficients by $\sqrt{2}$. This is a polynomial with irrational coefficients, which has the same roots as the original.2011-02-24
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    @J.J. What happens if they are different? (e.g. rt2, rt3)2018-08-16

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The roots of a polynomial with algebraic coefficients are all algebraic, and a monic polynomial whose roots are all algebraic has algebraic coefficients.

So a monic polynomial with some transcendental coefficient must have at least one transcendental root (and vice versa), but it can also have algebraic roots (for example, $0$ is a non transcendental root of $X^2- \pi X = 0$).

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    If you change $\sqrt{2}$ to $\pi$ in J.J.'s comment you need not have a transcendental root even if the coefficients are transcendental.2011-02-24
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    right, I forgot to keep the monic condition.2011-02-24