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Previous: Case 1: p 
We may assume that Z = p^{k}Z_{0} and
(p, X, Y, Z_{0}) are mutually
coprime. By writing
p = (unit)^{ . } in the ring
R, we obtain an equation of the form
U^{p} +
V^{p} + (unit)
W^{p} = 0 with
m > 0
where (U, V, W) are in R so that
(U, V, W,) are mutually
coprime. 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 coprime to p and thus as well, we
see that
(U + V)/ have distinct residue classes modulo
R. But then, by the pigeonhole 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 
= 
a_{0} 

U + V 
= 
a_{k}; fork > 0 

where all the a_{k} are elements of R that are coprime 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 I_{j} in R such that
I_{j}^{p} = a_{j}R. Assume that I_{0},
I_{1} and I_{p  1} are principal, then we have the equations
U + V 
= 
^{ . }u^{ . }b_{0}^{p} 

U + V 
= 
^{ . }v^{ . }b_{1}^{p} 

U + V 
= 
^{ . }w^{ . }b_{1}^{p} 

for some units u, v and w in R and some elements b_{0},
b_{1} and b_{1} in R. Eliminating U and V from these
equations we obtain
^{ . }u^{ . }b_{0}^{p} 
^{ . }v^{ . }b_{1}^{p} =
(
^{ . }w^{ . }b_{1}^{p} 
^{ . }u^{ . }b_{0}^{p})
which becomes
b_{1}^{p} +
v_{1}^{ . }b_{1}^{p} +
^{ . }v_{2}b_{0}^{p} = 0
where v_{1} and v_{2} are units (we use here the fact that 1 +
is a unit in R). Modulo pR the last term on the lefthand side
vanishes since
l p > (p  1). Thus we see that v_{1} is congruent
to a pth power and thus an integer modulo pR. By section 1 we
have a representation of Galois as required, unless v_{1} is a pth
power. If v_{1} = v_{3}^{p}, then
(U, V, W) = (b_{1}, v_{3}b_{1}, b_{0}) satisfy
U^{p} +
V^{p} + (unit)
W^{p} = 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 I_{0}, I_{1}, I_{p  1} is nonprincipal. 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
20021122