Hilberts tenth problem over $\mathbb R$ with coefficients in $\mathbb Q$

Question

Consider the following decision problem:

Given: An equation $f(x_1, \dots, x_n) = 0$ where $f(x_1, \dots, x_n)$ is a polynomial with variables $x_1, \dots, x_n$ and coefficients in $\mathbb Z$.

To decide: Is this equation solvable in $\mathbb R$?

It is well-known that Tarski proved that this decision problem is decidable. I wonder whether this problem is decidable when we allow the coefficients to be arbitrary rational numbers.

My question is: Is the following decision problem decidable?

Given: An equation $f(x_1, \dots, x_n) = 0$ where $f(x_1, \dots, x_n)$ is a polynomial with variables $x_1, \dots, x_n$ and coefficients in $\mathbb Q$.

To decide: Is this equation solvable in $\mathbb R$?

Answer 1

Hint: Given a polynomial $f$ with rational coefficients, we can mechanically produce a polynomial $g$ with integer coefficients such that the equation $f=0$ has a solution in the reals if and only if the equation $g=0$ has a solution in the reals.

Remark: On the other hand, if we seek solutions in $\mathbb{Q}$, then whether there is an algorithm is a long-standing open problem.