Let $x_1,\dots,x_n$ be complex numbers. Some authors define $x_1,\dots,x_n$ to be algebraically independent numbers if $x_1,\dots,x_n$ are algebraically independent over the field $\mathbb{Q}$ of rational numbers, while others call them algebraically independent numbers if they are algebraically independent over the field $\bar{\mathbb{Q}}$ of algebraic numbers.
I am pretty sure that the two definitions are equivalent, but I have a very elementary knowledge of algebra, so I cannot prove this equivalence, nor I could find some book which includes this statement. Could someone help me by providing me a proof or some reference? Thank you very very ... much for your help in advance.
This is a retelling of my comment.
Assume there is a nonzero polynomial $P$ with algebraic coefficients such that $P(x_1,\ldots,x_n)=0$.
There exists a finite Galois extension $K/\mathbb{Q}$ (with group $G$) containing all the coefficients of $P$.
Consider the nonzero polynomial $Q=\prod_{\sigma \in G}{\sigma(P)}$. It has coefficients in $K$ and is invariant under $G$ so has rational coefficients. Moreover, it’s clear $Q(x_1,\ldots,x_n)=0$.