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A new proof of a theorem of Kummer

1966
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Proceedings of the American Mathematical Society
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Let p be an odd prime and denote by K the field obtained by adjoining the pth roots of unity to Q, the rational numbers. Let f be a fixed primitive pth root of unity and set ir= 1 -f. The following theorem, due to Kummer, is of importance in proving the nonsolvability of xp+yp = z" in nonzero rational integers for regular primes p. Theorem. Let e be a unit in K and suppose that e = a (mod irp), where a is a rational integer. Then if p is regular there exists eiEK such that ep = e. The object of

doi:10.1090/s0002-9939-1966-0188195-9
fatcat:2vs4v67wrrazlch2v4vr7ec5vm