Math 676. Tame ramification and composite fields
1.
Review of tameness
Let
F
be the fraction field of a complete discrete valuation ring
A
with residue field
k
. Recall that a finite
separable extension
F
0
/F
(with valuation ring
A
0
and residue field
k
0
that are necessarily finite free modules
over
A
and
k
respectively) is
tamely ramified
if
k
0
/k
is separable and char(
k
)

e
(
F
0

F
). If moreover
k
0
=
k
then we say that
F
0
/F
is
totally tamely ramified
; that is,
F
0
/F
is totally tamely ramified if
e
(
F
0

F
) = [
F
0
:
F
]
and char(
k
)

e
(
F
0

F
).
If
π
∈
A
is a uniformizer and
e >
0 is a positive integer not divisible by char(
F
) then
X
e

π
∈
A
[
X
]
is separable and moreover irreducible over
F
(by Gauss’ Lemma and Eisenstein’s criterion).
Thus,
F
0
=
F
[
X
]
/
(
X
e

π
) is a field, typically denoted
F
(
π
1
/e
) (with
π
1
/e
denoting the residue class of
X
), and
F
0
/F
is separable of degree
e
with
π
1
/e
in the valuation ring contributing a factor of
e
= [
F
0
:
F
] in
e
(
F
0

F
). This
forces
F
0
/F
to be totally ramified with
π
1
/e
as a uniformizer, so via
π
1
/e
adic expansions we see that the
valuation ring of
F
0
is exactly
A
0
=
A
[
π
1
/e
] =
A
[
X
]
/
(
X
e

π
). This is a special case of the proof in class
that adjoining the root to any Eisenstein polynomial gives a uniformizer for a totally ramified extension.
If char(
k
)

e
then
F
(
π
1
/e
) is a totally tamely ramified extension, and we have seen in an earlier lecture
that every totally tamely ramified extension of
F
arises by this construction for a suitable
π
.
That is,
whereas we proved in general that every totally ramified extension of
F
is obtained by adjoining a root to
an Eisenstein polynomial over
A
, in the totally tame case we saw (via Hensel’s Lemma) that this Eisenstein
polynomial could be chosen to be of an especially simple form, namely
X
e

π
for
some
choice of uniformizer
π
in
A
.
Beware that for a general tame extension
F
0
/F
(with valuation ring
A
0
and residue
k
0
, but possibly
[
k
0
:
k
]
>
1), it is
not
usually the case that there is a uniformizer of
F
0
whose
e
(
F
0

F
)th power lies in
F
.
The best we can say is that if
F
u
/F
is the maximal unramified subextension then this has residue field
k
0
(here we use that
k
0
/k
is separable!) and
F
0
/F
u
is totally tamely ramified (why?). Hence,
F
0
=
F
u
(
π
1
/e
u
)
with
e
=
e
(
F
0

F
u
) =
e
(
F
0

F
u
)
e
(
F
u

F
) =
e
(
F
0

F
) and
π
u
∈
F
u
some uniformizer, but usually
π
u
cannot be
found inside of
F
. In Homework 11 there is given an explicit example of such impossibility for
F
=
Q
5
and
[
F
0
:
F
] = 4 with
e
(
F
0

F
) = 2.
To summarize, there is a reasonably “concrete” description of tamely ramified extensions
F
0
of
F
, espe
cially when
F
is a local field (in which case the maximal unramified subextension
F
u
in
F
0
/F
is a cyclotomic
extension
F
(
ζ
q
f

1
) with
q
= #
k
and
f
=
f
(
F
0

F
) = [
k
0
:
k
]). We would like to show that the property of
tameness is reasonably wellbehaved in the sense that it is preserved under formation of composite fields.
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 Fall '11
 Staff
 Math, Number Theory, residue ﬁeld, valuation ring, eth root

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