Suppose
is not Galois. Then , , and are defined for each prime
,
but we need not have
or
. We do still have that
, by the Chinese Remainder Theorem.
For example, let
. We know that
. Thus
, 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.

William Stein
2012-09-24