Number Theory: An Introduction to Mathematics

1 Valued Fields 263

(iv) LetF =K((t))be the field of all formal Laurent series f(t)=

∑

n∈Zαnt

n

with coefficientsαn∈Ksuch thatαn=0 for at most finitely manyn<0. A non-
archimedean absolutevalue is defined onFby putting, for a fixedq>1,

| 0 |= 0 , |f|=q−v(f) iff= 0 ,

wherev(f)is the least integernsuch thatαn=0.

(v) Let∑ F = Cζ((z))denote the field of all complex-valued functions f(z) =

n∈Zαn(z−ζ)

nwhich are meromorphic atζ∈C.Anyf∈Fwhich is not identically

zero can be uniquely expressed in the formf(z)=(z−ζ)vg(z),wherev=vζ(f)is
an integer,gis holomorphic atζandg(ζ)=0. A non-archimedean absolute value is
defined onFby putting, for a fixedq>1,

| 0 |ζ= 0 , |f|ζ=q−vζ(f) iff= 0.

It should be noted that in examples (iii) and (iv) the restriction of the absolute value
to the ground fieldKis the trivial absolute value, and the same holds in example (v)
for the restriction of the absolute value toC. For all the absolute values considered in
examples (iii)–(v) the value group is an infinite cyclic group.
We now derive some simple properties common to all absolute values. The
notation in the statement of the following lemma is a bit sloppy, since we use
the same symbol to denote the unit elements of bothFandR(as we have already
done for the zero elements).

Lemma 1In any field F with an absolute value||the following properties hold:

(i)| 1 |= 1 ,|− 1 |= 1 and, more generally,|a|= 1 for every a∈F which is a root
of unity;
(ii)|−a|=|a|for every a∈F;
(iii)||a|−|b||∞≤|a−b|for all a,b∈F, where||∞is the ordinary absolute value
onR;
(iv)|a−^1 |=|a|−^1 for every a∈F with a= 0.

Proof By takinga=b=1in(V2)and using(V1), we obtain| 1 |=1. Ifan=1for
some positive integern, it now follows from(V2)thatα=|a|satisfiesαn=1. Since
α>0, this impliesα=1. In particular,|− 1 |=1. Takingb=−1in(V2),wenow
obtain (ii).
Replacingabya−bin(V3), we obtain

|a|−|b|≤|a−b|.

Sinceaandbmay be interchanged, by (ii), this implies (iii). Finally, if we take
b=a−^1 in(V2)and use (i), we obtain (iv).

It follows from Lemma 1(i) that a finite field admits only the trivial absolute value.
We show next how non-archimedean and archimedean absolute values may be dis-
tinguished from one another. The notation in the statement of the following proposition
is very sloppy, since we use the same symbol to denote both the positive integernand
the sum 1+ 1 +···+1(nsummands), although the latter may be 0 if the field has
prime characteristic.