Real Nullstellensatz and sums of squares

In this paper we highlight the foundational principles of sums of squares in the study of Real Algebraic Geometry. To this aim the article is designed as mainly a self-contained presentation of a variation of the standard proof of Real Nullstellensatz, the only relevant omission being the (long) proof of the Tarski-Seidenberg theorem. On the way we see how the theory follows closely developments in algebra and model theory due to Artin and Schreier. This allows us to present on the way Artin’s solution to Hilbert’s 17th Problem: whether positive polynomials are sums of squares. These notes are intended to be accessible to math students of any level.