Real numbers that are not the roots of any polynomial equation with algebraic coefficients

Question

An algebraic number is a number which is a root of some non-zero polynomial equation with rational coefficients.

A transcendental number is a number which is not a root of any non-zero polynomial equation with rational coefficients.

Are there any real numbers that are not the roots of any polynomial equation with algebraic coefficients?

I'm guessing that somebody must have asked this before somewhere, though I have not been able to find it, so please excuse me for the possible duplicate...

Thanks

Answer 1

The complement of the transcendentals in $\mathbb{R}$ is the real algebraic numbers. The algebraic numbers are algebraically closed. Therefore every polynomial with non-transcendental coefficients has algebraic coefficients and thus algebraic roots.

Answer 2

"Are there real numbers ..." Yes, there are, and in fact those numbers coincide with the transcendental numbers. In fact, if $x$ is a root of a polynomial with non-transcendental(=algebraic) coefficients, then $x$ is itself algebraic.

Answer 3

Now that you've added non-trancendental (i.e. "algebraic") to the question, the answer is yes.

The roots of a polynomial with algebraic coefficients are also roots of some polynomial (possibly of higher degree) with rational coefficients.