A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

148CHAPTER 19. EXTENSIONS AND NORMALIZATIONS OF VALUATIONS
for a ∈ K, where σ1, . . .,σN ∈ Gal(M/K) extend the embeddings of L into M.
Hence
as required.
NormL/K(a) = NormL/K(a)
=
σn(a)
1≤n≤N
= a N ,
Corollary 19.1.6. Let w1, . . .,wN be a basis for L over K. Then there are positive
constants c1 and c2 such that
N
bnwn
c1 ≤
n=1
max{|bn| : n = 1, . . .,N}
≤ c2
for any b1, . . .,bN ∈ K not all 0.
Proof. For N n=1 bnwn
and max |bn| are two norms on L considered as a vector
space over K.
I don’t believe this proof, which I copied from Cassels’s article. My problem
with it is that the proof of Theorem 19.1.2 does not give that C ≤ 2, i.e., that the
triangle inequality holds for · . By changing the basis for L/K one can make any
nonzero vector a ∈ L have a0 = 1, so if we choose a such that |a| is very large,
then the ∆ in the proof will also be very large. One way to fix the corollary is to
only claim that there are positive constants c1, c2, c3, c4 such that
c1 ≤
N
c3
bnwn
n=1
max{|bn| c4 ≤ c2.
: n = 1, . . .,N}
Then choose c3, c4 such that · c3 and | · | c4 satisfies the triangle inequality, and
prove the modified corollary using the proof suggested by Cassels.
Corollary 19.1.7. A finite extension of a completely valued field K is complete
with respect to the extended valuation.
Proof. By the proceeding corollary it has the topology of a finite-dimensional vector
space over K. (The problem with the proof of the previous corollary is not an issue,
because we can replace the extended valuation by an inequivalent one that satisfies
the triangle inequality and induces the same topology.)
When K is no longer complete under | · | the position is more complicated: