In the introduction of his first paper on nonstandard analysis [Proc. K. Ned. Akad. Wet., Ser. A 64, 432-440 (1961; Zbl 0102.007)], A. Robinson refers to the work of Artin and Schreier on the algebraic construction of real fields. This paper analyzes the relations between this paper of Artin-Schreier and the genesis of Robinson’s nonstandard analysis, specially through a study of Robinson’s first book on the metamathematics of algebra. There, with his own methods, Robinson reproves the possibility of a non-archimedian ordered field where the basic theorems of real algebra (in particular the Bolzano theorem) are satisfied. The author concludes by claiming that Robinson’s nonstandard analysis is a consequence of progress made outside of analysis, namely Skolem’s extensions of the system of natural numbers and the concept of real closed field.