Is there an algorithm to decide whether an algebraic system 
\begin{gathered}
 {f\_1}({x\_1}, \ldots ,{x\_n}) = 0 \hfill \\
 \vdots \hfill \\
 {f\_m}({x\_1}, \ldots ,{x\_n}) = 0 \hfill \\ 
\end{gathered}
where $f\_1,\ldots,f\_m$ are polynomials with given rational coefficients, has a solution in real numbers?



Decidability of an Algebraic System in Real Numbers
Yes, this follows from Tarski's theorem (1951) that the first order theory of real closed fields admits elimination of quantifiers. See also the [Tarski-Seidenberg theorem](https://en.wikipedia.org/wiki/Tarski%E2%80%93Seidenberg_theorem).


**P.S.** A consequence of these theorems is that if the given polynomial system has a solution in real numbers, then it also has a solution in real algebraic numbers.


