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2005/10/02 | 阿廷映射[Artin Map]
类别(∑〖数学〗)
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发表于 10:21
Take
a
number field
and
an
Abelian extension
, then form a prime divisor that is divided by all ramified primes of the extension
. Now define a map
from the fractional ideals relatively prime to
to the
Galois group
of
that sends an ideal
to
. This map is called the Artin map. Its importance lies in the kernel, which
Artin's reciprocity theorem
states contains all fractional ideals that are only composed of primes that split completely in the extension
.
This is the reason that it is a reciprocity law. The inertia degree of a prime can now be computed since the smallest exponent
for which
belongs to this kernel, which is exactly the inertia degree, is now known. Now because
is unramified and
is Galois,
, with
the inertia degree and
the number of factors into which
splits when it is extended to
. So it is completely known how
behaves when it is extended to
.
This is completely analogous to quadratic reciprocity because it also determines when an unramified prime
splits (
,
) or is inert (
,
). Of course, quadratic reciprocity is much simpler since there are only two possibilities in this case.
In the special case of the
Hilbert class field
, this kernel coincides with the ordinary class group of the extension.
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