A new proof of a theorem of Kummer
Proceedings of the American Mathematical Society
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
... = e. The object of this note is to give a new proof of this theorem. The newness lies in the proof of the following theorem, from which Rummer's theorem is easily derived. In the statement of the theorem and throughout the cohomology groups in question are the Tate cohomology groups (see [3, Chapter VIII]). Theorem. Let Ebea number field and L a cyclic extension of E of odd prime degree. Denote by U the group of units in L and by G the Galois group ofL/E. Then H~l(U, G)9*0. Proof. Let (E/Q)=rA-2s where r is the number of real infinite primes of E and 5 is the number of complex infinite primes. Thus if V is the group of units of E, Fis of rank t = rArS -1. Let be an isomorphism of E into the complex numbers. Then can be extended in exactly p = (L/E) ways to L. Ii d>(E) is real then any extension of <p to L must also be real since p is odd. (If not, then the image of L would be of degree 2 over its maximal real subfield implying that p is even.) Thus (L/Q)=prAr2ps and L has pr real infinite primes and p5 complex infinite primes. Therefore U is of rank u = prA-ps-l=ptA-p-l.