TITLE: Roots of a polynomial equation with algebraic coefficient algebraic?
QUESTION [5 upvotes]: Why is it that the roots of a polynomial with algebraic coefficients is algebraic? Is it because you can rewrite the polynomial $f(x)=\sum_{i=1}^n a_ix^i = \sum_{i=1}^n x^i(a_i) = 0?$ where the last equality holds by definition of $a_i$ being algebraic.

REPLY [7 votes]: It is because $\alpha$ is algebraic over $K$ if and only if $K(\alpha)$ has finite dimension as a $K$-vector space. Hence if $\alpha$ is a root of $p(x)=\sum_{i=1}^n a_ix^i$, and the $a_i$s are algebraic over $K$, by a trivial induction, $L=K(a_1,\dots,a_n)\,$ has finite dimension over K, and $M=L(\alpha)$ has  finite dimension over $L$. Thus $\alpha$ is algebraic over $K$ since:
$$\dim_KM=\dim_LM\cdot \dim_K L$$