TITLE: Decidability of an Algebraic System in Real Numbers
QUESTION [8 upvotes]: 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?

REPLY [8 votes]: 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.
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.