is not Galois. Then , , and are defined for each prime
but we need not have
. We do still have that
, by the Chinese Remainder Theorem.
For example, let
. We know that
, so for we have and .
Working modulo we have
and the quadratic factor is irreducible. Thus
Thus here , , , and .
Thus when is not Galois we need not have that the
are all equal.