A Block-theoretic Proof of Burnside's Normal $p$-complement Theorem

In [3, Theorem 6.7B], the authors use the Main Theorems of Brauer to give a proof of Burnside's Normal $p$-complement Theorem. Unfortunately, the proof contains an error. We take this opportunity to give a proof along similar lines, circumventing the error by means of a well-known result on traces of totally positive cyclotomic integers.