Are all hyper-algebraic numbers algebraic?

Question

I know an algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).
So Let hyper-algebraic number be any complex number that is a root of a non-zero polynomial in one variable with algebraic coefficients

My question : are all hyper-algebraic numbers algebraic ?

---

for example :
let define $x_0$ as the root of
$x^5+x=10$ something like $x_0≈1.53301...$

I wonder :

if $y_0$ is any root of $y^5+x_0^2y=(1+{\sqrt x_0})$ (this polynomial has algebraic coefficients)
then is $y_0$ algebraic ?
I mean is there any polynomial with integer coefficients (may be we cannot find it , but it exists ) that $y_0$ is its root ?

Answer 1

The field of algebraic numbers $\overline{\mathbb{Q}}$ is algebraically closed. Therefore every root of a polynomial with algebraic coefficients is also algebraic.

Reference: Corollary $1.4$ here.