Number Theory: An Introduction to Mathematics

274 VI Hensel’sp-adic Numbers

It also follows at once from Lemma 8 that if a sequence(an)of elements ofF

converges to a limita=0, then|an|=|a|for all largen. Hence the value group of

the fieldFis the same as the value group of its completionF ̄. The next lemma has an

especially appealing corollary.

Lemma 10Let F be a field with a non-archimedean absolute value||.Thena

sequence (an) of elements of F is a fundamental sequence if and only if

limn→∞|an+ 1 −an|= 0.

Proof If|an+ 1 −an|→0, then for eachε>0 there is a corresponding positive

integerN=N(ε)such that

|an+ 1 −an|<ε forn≥N.

For any integerk>1,

an+k−an=(an+ 1 −an)+(an+ 2 −an+ 1 )+···+(an+k−an+k− 1 )

and hence

|an+k−an|≤max{|an+ 1 −an|,|an+ 2 −an+ 1 |,...,|an+k−an+k− 1 |}<εforn≥N.

Thus(an)is a fundamental sequence. The converse follows at once from the definition

of a fundamental sequence. 

Corollary 11In a field F with a complete non-archimedean absolute value||,an

infinite series

∑∞

n= 1 anof elements of F is convergent if and only if|an|→^0.

LetFbe a field with a nontrivial non-archimedean absolute value||and put

R={a∈F:|a|≤ 1 },

M={a∈F:|a|< 1 },

U={a∈F:|a|= 1 }.

ThenRis the union of the disjoint nonempty subsetsMandU. It follows from the de-
finition of a non-archimedeanabsolute value thatRis a (commutative) ring containing
the unit element ofFand that, for any nonzeroa∈F, eithera∈Rora−^1 ∈R(or
both). MoreoverMis an ideal ofRandUis a multiplicative group, consisting of all
a∈Rsuch that alsoa−^1 ∈R. Thus a proper ideal ofRcannot contain an element of
Uand henceMis the unique maximal ideal ofR. Consequently (see again Chapter I,
§8) the quotientR/Mis a field.
We callRthevaluation ring,Mthevaluation ideal,andR/Mtheresidue fieldof
the valued fieldF.
We draw attention to the fact that the ‘closed unit ball’Ris both open and closed
in the topology induced by the absolute value. For ifa∈Rand|b−a|<1, then also
b∈R.Furthermore,ifan∈Randan→athena∈R,since|an|=|a|for all largen.
Similarly, the ‘open unit ball’Mis also both open and closed.
In particular, letF=Qbe the field of rational numbers and||=||pthep-adic
absolute value. In this case the valuation ringR=Rpis the set of all rational numbers