David W. answered 06/03/19
Music Theory Professor and Composer with 5+ Years Teaching Experience
Major and minor are actually two modes (Ionian and Aeolian). They took over music from the other modes in the early Baroque, and key signatures were developed to notate them efficiently.
So if we're in C Mixolydian (which has the same accidentals as F major, btw, not F minor)—we're in C Mixolydian, not C major or C minor. The question then becomes what key signature to use, and there is no universally accepted answer. Some composers use the equivalent major or minor key signature (in this case it would be one flat, as in Fmajor). Some composers use the key signature of the most closely related major or minor scale with the same tonic (in this case, C major). You can find both.