I guess you should definitely take a look at
WAERDEN, B. L. van der, "How the proof of Baudet's conjecture was found", Studies in Pure Mathematics (papers presented to R. Rado on the occassion of his 65th birthday, ed. by L. Mirsky, Academic Press), pp. 252-260.
According to N. G. De Bruijn, the note by B. L. van der Waerden
... was partly intendend as an illustration of the author's ideas on the psychology of mathematical invention. The reading of the report is recommended to all those for whom understanding is not just formal verification, but rather a procedure by which intuitive ideas and experiences are linked together to each other in order to build up the final mathematical structure... The reading of van der Waerden's report is also recommended to those who are interested to learn about the discussion (with Artin and Schreier) that preceded van der Waerden's discovery, and to learn about the actual contributions of Artin and Schreier to the solution of the problem.
It is important to add that Baudet's conjecture goes now under the name of van der Waerden's theorem on arithmetic progressions. The proof of this cute result is showcased in A. Y. Khinchin's Three Pearls of Number Theory. Yet, be warned that Khinchin does not present the original approach by van der Waerden, but an attack communicated to him by a M. A. Lukomskaya (cf. my reply to thisthis discussion).