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## Case 2: p| XYZ

We may assume that Z = pkZ0 and (p, X, Y, Z0) are mutually co-prime. By writing p = (unit) . in the ring R, we obtain an equation of the form

Up + Vp + (unit)Wp = 0 withm > 0

where (U, V, W) are in R so that (U, V, W,) are mutually co-prime. Let (U, V, W) be a collection of elements of R that satisfy such an equation with m the least possible. Then divides one of the factors (U + V). But then we have

(U + V) - (U + V) = (1 - )V = (unit) . V

and thus, divides all the factors (U + V). Moreover, since V is co-prime to p and thus as well, we see that (U + V)/ have distinct residue classes modulo R. But then, by the pigeon-hole principle there is at least one 0 j (p - 1) such that (U + V) is divisible by in R. Replacing V by V we may assume that (U + V) is divisible by for some l > 1. Hence we may write
 U + V = a0 U + V = ak; fork > 0

where all the ak are elements of R that are co-prime to and with each other (as in the previous case). This gives us the identity l + (p - 1) = mp or equivalently l = (m - 1)p + 1. Since l 2 we have m 2. Now by unique factorisation of ideals in R we see that there are ideals Ij in R such that Ijp = ajR. Assume that I0, I1 and Ip - 1 are principal, then we have the equations
 U + V = . u . b0p U + V = . v . b1p U + V = . w . b-1p

for some units u, v and w in R and some elements b0, b1 and b-1 in R. Eliminating U and V from these equations we obtain

. u . b0p - . v . b1p = ( . w . b-1p - . u . b0p)

which becomes

b1p + v1 . b-1p + . v2b0p = 0

where v1 and v2 are units (we use here the fact that 1 + is a unit in R). Modulo pR the last term on the left-hand side vanishes since l p > (p - 1). Thus we see that v1 is congruent to a p-th power and thus an integer modulo pR. By section 1 we have a representation of Galois as required, unless v1 is a p-th power. If v1 = v3p, then (U, V, W) = (b1, v3b-1, b0) satisfy

Up + Vp + (unit)Wp = 0

which contradicts the minimality of m since we have seen that m 2. Thus, either we have constructed a cyclic extension of the required type or one of I0, I1, Ip - 1 is non-principal. But then again by the principal result of Class Field theory we have a cyclic extension as required.

Next: 3 Transcendental computation of Up: 2 Construction of cyclic Previous: Case 1: p |
Kapil Hari Paranjape 2002-11-22