262 VI Hensel’sp-adic Numbers
A non-archimedean absolute value is indeed an absolute value, since(V1)implies that
(V3)′is a strengthening of(V3). An absolute value is said to bearchimedeanif it is
not non-archimedean.
The inequality(V3)is usually referred to as thetriangle inequalityand(V3)′as
the ‘strong triangle’, orultrametric, inequality.
IfFis a field with an absolute value||, then the set of real numbers|a|for all
nonzeroa∈Fis clearly a subgroup of the multiplicative group of positive real num-
bers. This subgroup will be called thevalue groupof the valued field.
Here are some examples to illustrate these definitions:
(i) An arbitrary fieldFhas atrivialnon-archimedean absolute value defined by
| 0 |= 0 , |a|=1ifa= 0.(ii) The ordinary absolute value
|a|=a ifa≥ 0 , |a|=−a ifa< 0 ,defines an archimedean absolute value on the fieldQof rational numbers. We will
denote this absolute value by||∞to avoid confusion with other absolute values onQ
which will now be defined.
If pis a fixed prime, any rational numbera =0 can be uniquely expressed
in the forma =epvm/n,wheree=±1,v =vp(a)is an integer andm,nare
relatively prime positive integers which are not divisible byp. It is easily verified that
a non-archimedean absolute value is defined onQby putting
| 0 |p= 0 , |a|p=p−vp(a) ifa= 0.We call this thep-adic absolute value.
(iii) LetF =K(t)be the field of all rational functions in one indeterminate with
coefficients from some fieldK. Any rational function f = 0 can be uniquely
expressed in the form f =g/h,wheregandhare relatively prime polynomials
with coefficients fromKandhismonic(i.e., has leading coefficient 1). If we denote
the degrees ofgandhby∂(g)and∂(h), then a non-archimedean absolute value is
defined onFby putting, for a fixedq>1,
| 0 |∞= 0 , |f|∞=q∂(g)−∂(h) iff= 0.Other absolute values onFcan be defined in the following way. Ifp∈ K[t]
is a fixed irreducible polynomial, then any rational functionf =0 can be uniquely
expressed in the form f = pvg/h,wherev =vp(f)is an integer,gandhare
relatively prime polynomials with coefficients fromKwhich are not divisible byp,
andhis monic. It is easily verified that a non-archimedean absolute value is defined
onFby putting, for a fixedq>1,
| 0 |p= 0 , |f|p=q−∂(p)vp(f) iff= 0.