Emil Artin's solution of Hilbert's 17th problem which asked whether every positive polynomial in any number of variables is a sum of squares of rational functions.

Artin's proof goes roughly as follows. If $p \in \mathbb R[x_1,\dots,x_n]$ it not a sum of squares of rational functions, then there is some real-algebraically closed extension $L$ of the field of rational functions in which $p$ is negative with respect to some total ordering (compatible with the field operations), i.e. there exists a $L$-point of $R[x_1,\dots,x_n]$ at which $p$ is negative. However, using a model theoretic argument, since $\mathbb R$ is also a real-closed field with a total ordering, there also has to be a real point such that $p<0$, i.e. there exists $x \in \mathbb R^n$ such that $p(x)< 0$. Hence, if $p$ is everywhere positive, then it is a sum of squares of rational functions.

The ingenius part is the use of a model theoretic argument and the bravery to consider a totally ordered real-algebraic closed extension of the field of rational functions.