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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 $ \geq$ 5 leads to a non-trivial representation

$\displaystyle \rho$ : Gal($\displaystyle \overline{K}$/K)$\displaystyle \to$$\displaystyle \Bbb$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 $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R $ \hookrightarrow$ K $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R from which we obtain the isomorphism

(K $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R)*1/(L $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R)*1 = ($\displaystyle \Bbb$C*1/$\displaystyle \Bbb$R*1)(p - 1)/2

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

$\displaystyle \nu$((K $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R)*1/U) = $\displaystyle {\frac{1}{p}}$ . $\displaystyle \nu$($\displaystyle \Bbb$C*1/$\displaystyle \Bbb$R*1)(p - 1)/2 . $\displaystyle \nu$((L $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R)*1/U+)

The formula for computing discriminants yields

$\displaystyle \mu$(K $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R/R) = $\displaystyle \mu$(L $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$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-

$\displaystyle {\frac{h_{-} \cdot
\nu({\Bbb C}^{*}_1 / {\Bbb R}^{*}_1)^{(p-1)/2}
}{p^{3/2} \cdot \mu(L\otimes _{{\Bbb Q}}{\Bbb R}/S)
}}$ = $\displaystyle \prod_{\chi(-1)=-1}^{}$L(1,$\displaystyle \chi$)

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 $ \nu$((L $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R)*1/U+) is difficult to compute. However, we have the subgroup U+ , cycl = U+ $ \cap$ Ucycl and one can compute $ \nu$((L $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R)*1/U+ , cycl). In fact one shows that

$\displaystyle \nu$((L $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R)*1/U+ , cycl) = $\displaystyle \mu$(L $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R/S) . $\displaystyle \prod_{\chi\text{ even }}^{}$L(1,$\displaystyle \chi$)

where the product runs over all non-trivial characters $ \chi$ such that $ \chi$(- 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 $ \in$ Ucycl such that v = wp then one shows easily that u is itself a cyclotomic unit. Let Q denote the quotient group (S/pS)*/($ \Bbb$Z/p$ \Bbb$Z)*. We obtain a natural homomorphism

m : Ucycl $\displaystyle \otimes$ ($\displaystyle \Bbb$Z/p$\displaystyle \Bbb$Z)$\displaystyle \to$Q

which is represented by a square matrix with entries from $ \Bbb$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) $ \cong$ 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 up previous
Next: Bibliography Up: Kummer's proof of Fermat's Previous: 3 Transcendental computation of
Kapil Hari Paranjape 2002-11-22