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SOLUTION TO MATH 8410, EXERCISE 2.4
PETE L. CLARK
Prerequisites
: This exercise uses Exercise 2.2 and Exercise 2.3.
Statement
: Let (
K,
 
) be a nonArchimedean normed ﬁeld, and let
L/K
be
a purely transcendental extension. Show that
 
extends to a norm on
L
.
Background
: We will review the deﬁnition of a purely transcendental extension.
More signiﬁcantly, we will prove the result by
transﬁnite induction
, so let’s begin
with a refresher
1
on that.
Recall that a
wellordered set
is a set
S
endowed with a total ordering rela
tion
≤
(a reﬂexive, antisymmetric, transitive relation such that for all
x, y
∈
S
, at
least one of
x
≤
y
and
y
≤
x
holds) which has the additional property that every
nonempty subset
T
of
S
has a least element. For instance, the natural numbers
N
with their usual ordering are a wellordered set, as are
N
∪ {∞}
, the natural
numbers with an additional element
∞
such that
x <
∞
for all
x
∈
N
. In fact,
for any wellordered set
S
, it turns out to be useful to deﬁne the wellordered set
S
+
=
S
∪{∞}
in the same way. (In particular,
S
+
has a maximal element, whereas
S
need not.)
The empty set
is
wellordered. Any nonempty wellordered set has a least ele
ment, which we might as well call 0.
2
If
S
is wellordered, and
x
is any element of
S
except possibly the maximal element (if any), then the set
{
y
∈
S

x < y
}
is
nonempty hence has a least element, the
successor
of
x
. In the natural numbers,
the successor of
n
is nothing else than
n
+1, so we may as well denote the successor
of
x
by
x
+ 1 in the general case. An element
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 Fall '11
 Staff
 Math, Number Theory

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