The structure of (complete) discretely valued fields. Today this is usually done with Teichmüller representatives and Witt vector machinery (e.g. Serre, Local Fields, ch. II §§4-6). But the main results had been proved with more complicated methods by H. Hasse and F. K. Schmidt (Crelle 170 (1934): 60 pages) before these tools were developed. Also, what is today called Artin-Schreier-Witt theory was developed by H. L. Schmid with very messy computations: These inspired Witt to invent his vectors, which then gave it a much more conceptual background. Cf. the "history" part of my answer here.
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