Number Theory: An Introduction to Mathematics

3 Completions 271

It is easily seen thatF ̄ is uniquely determined, up to an isomorphism which
preserves the absolute value. The fieldF ̄is called thecompletionof the valued fieldF.
The density ofFinF ̄implies that the absolute value on the completionF ̄is non-
archimedean or archimedean according as the absolute value onFis non-archimedean
or archimedean.
It is easy to see that in example (iv) of§1 the valued fieldF=K((t))of all formal
Laurent series is complete, i.e. it is its own completion. For let{f(k)}be a fundamental
sequence inF. Given any positive integerN, there is a positive integerM=M(N)
such that|f(k)−f(j)|<q−Nforj,k≥M. Thus we can write

f(k)(t)=

∑

n≤N

αntn+

∑

n>N

α(nk)tn for allk≥M.

Iff(t)=

∑

n∈Zαnt

n, then limk→∞|f(k)−f|=0.

On the other hand, given anyf(t)=

∑

n∈Zαnt

n∈K((t)),wehave|f(k)−f|→ 0

ask→∞,wheref(k)(t)=

∑

n≤kαnt

n∈K(t). It follows thatK((t))is the com-

pletion of the fieldK(t)of rational functions considered in example (iii) of§1, with
the absolute value||tcorresponding to the irreducible polynomialp(t)=t(for which
∂(p)=1).
The completion of the rational fieldQwith respect to thep-adic absolute value||p
will be denoted byQp, and the elements ofQpwill be calledp-adic numbers.
The completion of the rational fieldQwith respect to the ordinary absolute value
||∞is of course the real fieldR.In§6 we will show that the only fields with a com-
plete archimedean absolute value are the real fieldRand the complex fieldC,andthe
absolute value has the form||ρ∞for someρ>0. In factρ≤1, since 2ρ≤ 1 ρ+ 1 ρ=2.
Thus an arbitrary archimedean valued field is equivalent to a subfield ofCwith the
usual absolute value. (Hence, for a field with an archimedean absolute value||,|n|> 1
for every integern>1and|n|→∞asn→∞.) Since this case may be considered
well-known, we will in the following devote our attention primarily to the peculiarities
of non-archimedean valued fields.
We will later be concerned with extending an absolute value on a fieldFto a field
Ewhich is a finite extension ofF. Since all that matters for some purposes is thatE
is a vector space overF, it is useful to introduce the following definition.
LetFbe a field with an absolute value||and letEbe a vector space overF.
AnormonEis a map‖‖:E→Rwith the following properties:

(i)‖a‖>0foreverya∈Ewitha=0;
(ii)‖αa‖=|α|‖a‖for allα∈Fanda∈E;
(iii)‖a+b‖≤‖a‖+‖b‖for alla,b∈E.

It follows from (ii) that‖O‖=0. We will require only one result about normed vector
spaces:

Lemma 7Let F be a complete valued field and let E be a finite-dimensional vector
space over F. If‖‖ 1 and‖‖ 2 are both norms on E , then there exist positive constants
σ,μsuch that

σ‖a‖ 1 ≤‖a‖ 2 ≤μ‖a‖ 1 for every a∈E.