TITLE: Algebraic numbers and their closure
QUESTION [6 upvotes]: Are all of the roots/zeroes of a polynomial of finite degree with algebraic coefficients algebraic?
How about for a generalization of a polynomial wherein the indefinite is exponentiated to an algebraic power and then multiplied by an algebraic coefficient, where the number of terms is finite, provided that such a zero/root/value exists? In other words, must the values $z$ for which sums of summands of the form $az^b$, where $a$ and $b$ are algebraic, evaluate to zero be algebraic (if such $z$ exists)?

REPLY [5 votes]: I'll post my comments as an answer to reduce the number of unanswered questions.
The first question: Yes, all zeros to polynomials of finite degree with algebraic coefficients are algebraic, i.e. the field of algebraic numbers is algebraically closed. A proof sketch of this fact can be found here. 
A counter-example of the second question would be the equation $az^b-1=0$, where $a=b=\frac1{\sqrt2}$, which are algebraic, while its root $z=\sqrt2^{\sqrt2}$ is not.