Math 676. Some examples of extension fields
The purpose of this handout is to probe the hypotheses of some of our results in class concerning the
structure of finite extensions of local fields, or more generally of fraction fields of complete discrete valuation
rings.
1.
Number of extensions of a local field
In class we saw that if
K
is a local field and
n
is a positive integer not divisible by char(
K
) then the set
of
K
isomorphism classes of degree
n
extensions of
K
is a finite set. Recall that the condition char(
K
)

n
is crucial in the proof, as otherwise the compact space of Eisenstein polynomials over
K
with degree
n/f
(for a fixed
f

n
) contains some inseparable polynomials (if char(
K
)

(
n/f
)) and so the locus of separable
Eisenstein polynomials of degree
n/f
would appear to be probably noncompact (as it is the complement of
a “hypersurface” in a compact space, the “hypersurface” being the zerolocus of the universal discriminant
for monic polynomials of degree
n/f
).
Let us show that this apparent noncompactness for char(
K
)

n
is genuine and in fact leads to coun
terexamples to the desired finiteness statement in such cases. More specifically, we fix a local field
K
with
characteristic
p >
0 and we seek to show that there are infinitely many totally ramified separable
p
extensions
of
K
(up to
K
isomorphism). Iterating such a construction would give infinitely many totally ramified sep
arable
p
r
extensions (up to
K
isomorphism) for any
r >
0, and then forming (linearly disjoint) composites
with a totally tame extension
K
(
π
1
/e
0
) for any
e
0
>
0 not divisible by
p
(and for a fixed choice of uniformizer
π
∈
K
) would give rise to infinitely many totally ramified separable extensions of
K
(up to
K
isomorphism)
with any desired degree
n
divisible by
p
.
To carry out the construction of the infinitely many nonisomorphic totally ramified separable
p
extensions
of
K
, note that we do not really need to keep track of ramification: a separable
p
extension
K
0
/K
either
has
e
(
K
0

K
) =
p
or
e
(
K
0

K
) = 1. In the first case
K
0
/K
is totally ramified, and in the second case it is an
unramified
p
extension. There is only one unramified extension of each degree (due to the Galois theory of
finite fields), and removing one element from an infinite list still yields an infinite set. Thus, we focus on the
following problem: construct infinitely many separable
p
extensions
K
0
/K
that are pairwise nonisomorphic
over
K
. (Note that it is crucial to emphasize the
K
isomorphism aspect, since if we ignore the
K
structure
and consider mere abstract topological field isomorphism then everything will have to collapse: any local
field is abstractly topologically isomorphic to a formal Laurent series field over its residue field and so is
determined up to abstract (topological) field isomorphism by its residue field alone.)
We will make examples that are even Galois (cyclic) of degree
p
, via Artin–Schreier theory. Recall the
main result of Artin–Schreier theory: for any field
F
with characteristic
p >
0, the set of
F
isomorphism
classes of cyclic
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 Fall '11
 Staff
 Math, Number Theory, UK, Discrete valuation ring, residue ﬁeld, complete discrete valuation

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