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Next: 3 Transcendental computation of Up: 2 Construction of cyclic Previous: Case 1: p |

Case 2: p| XYZ

We may assume that Z = pkZ0 and (p, X, Y, Z0) are mutually co-prime. By writing p = (unit) . $ \lambda^{(p-1)}_{}$ in the ring R, we obtain an equation of the form

Up + Vp + (unit)$\displaystyle \lambda^{mp}_{}$Wp = 0 withm > 0

where (U, V, W) are in R so that (U, V, W,$ \lambda$) 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 $ \lambda$ divides one of the factors (U + $ \omega^{j}_{}$V). But then we have

(U + $\displaystyle \omega^{j}_{}$V) - (U + $\displaystyle \omega^{k}_{}$V) = $\displaystyle \omega^{j}_{}$(1 - $\displaystyle \omega^{k-j}_{}$)V = (unit) . $\displaystyle \lambda$V

and thus, $ \lambda$ divides all the factors (U + $ \omega^{j}_{}$V). Moreover, since V is co-prime to p and thus $ \lambda$ as well, we see that (U + $ \omega^{j}_{}$V)/$ \lambda$ have distinct residue classes modulo $ \lambda$R. But then, by the pigeon-hole principle there is at least one 0 $ \leq$ j $ \leq$ (p - 1) such that (U + $ \omega^{j}_{}$V) is divisible by $ \lambda^{2}_{}$ in R. Replacing V by $ \omega^{j}_{}$V we may assume that (U + V) is divisible by $ \lambda^{l}_{}$ for some l > 1. Hence we may write
U + V = $\displaystyle \lambda^{l}_{}$a0  
U + $\displaystyle \omega^{k}_{}$V = $\displaystyle \lambda$ak; fork > 0  

where all the ak are elements of R that are co-prime to $ \lambda$ 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 $ \geq$ 2 we have m $ \geq$ 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 = $\displaystyle \lambda^{l}_{}$ . u . b0p  
U + $\displaystyle \omega$V = $\displaystyle \lambda$ . v . b1p  
U + $\displaystyle \omega^{-1}_{}$V = $\displaystyle \lambda$ . 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

$\displaystyle \lambda^{l}_{}$ . u . b0p - $\displaystyle \lambda$ . v . b1p = $\displaystyle \omega$($\displaystyle \lambda$ . w . b-1p - $\displaystyle \lambda^{l}_{}$ . u . b0p)

which becomes

b1p + v1 . b-1p + $\displaystyle \lambda^{l-1}_{}$ . v2b0p = 0

where v1 and v2 are units (we use here the fact that 1 + $ \omega$ is a unit in R). Modulo pR the last term on the left-hand side vanishes since l $ \geq$ 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)$\displaystyle \lambda^{(m-1)p}_{}$Wp = 0

which contradicts the minimality of m since we have seen that m $ \geq$ 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.
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Next: 3 Transcendental computation of Up: 2 Construction of cyclic Previous: Case 1: p |
Kapil Hari Paranjape 2002-11-22