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# 4 Divisiblity of the Class number by p

Combining the results of sections 1 and 2 we have shown that any counter-example to Fermat's Last theorem for a prime p 5 leads to a non-trivial representation

: Gal(/K)Fp

which is unramified everywhere; here K denotes the subfield of complex numbers generated by the p-th roots of unity. Kummer called primes which admit such representations irregular. He showed that there are indeed such primes (p = 37 is one such) and hence this particular attempt to prove Fermat's Last theorem fails. We now wish to show how one goes about checking whether a prime is irregular. We apply the results of Section 3 in the special case where K is the prime cyclotomic field of section 1 and also to the totally real subfield L. First of all we use the divisibility of the class number h of R by the class number h+ of S to write h = h+ . h- for some integer h-. Let W denote the (finite cyclic) group of roots of unity in K. Then we have U = W . U+, where U+ denotes the group of units in S and so #(U/U+) = #(W/{±1}) = p. We have the natural inclusion L R K R from which we obtain the isomorphism

(K R)*1/(L R)*1 = (C*1/R*1)(p - 1)/2

since (p - 1)/2 is the degree of L over Q. From this we deduce that

((K R)*1/U) = . (C*1/R*1)(p - 1)/2 . ((L R)*1/U+)

The formula for computing discriminants yields

(K R/R) = (L R/S)2 . p1/2

since p is the norm of the relative discrimant. Thus the class number formulas for K and L then give a formula for h-

= L(1,)

Hence h- can be computed explicitly and in closed form. In particular, the divisibility of h- by p is an easily computable criterion. The divisibility of h+ by p is more complicated. As remarked earlier, the term ((L R)*1/U+) is difficult to compute. However, we have the subgroup U+ , cycl = U+ Ucycl and one can compute ((L R)*1/U+ , cycl). In fact one shows that

((L R)*1/U+ , cycl) = (L R/S) . L(1,)

where the product runs over all non-trivial characters such that (- 1) = 1. The class number formula for h+ becomes

h+ = [U+ : U+ , cycl] = [U : Ucycl].

This is the first coincidence that makes Kummer's calculations possible. From the above identity we see that if p divides h+ then we have a real unit u such that its p-th power is a cyclotomic unit but u is not itself cyclotomic. Hence v = up is a cyclotomic unit which is congruent to an integer modulo pS. If we find a w Ucycl such that v = wp then one shows easily that u is itself a cyclotomic unit. Let Q denote the quotient group (S/pS)*/(Z/pZ)*. We obtain a natural homomorphism

m : Ucycl (Z/pZ)Q

which is represented by a square matrix with entries from Fp. The preceding remarks imply that p| h+ only if det(m) = 0. The second coincidence that makes Kummer's calculation work is that det(m) h-(mod p). Thus we see that p| h if and only if p| h-. Hence we can easily check which primes are regular. to3em

Next: Bibliography Up: Kummer's proof of Fermat's Previous: 3 Transcendental computation of
Kapil Hari Paranjape 2002-11-22