Number Theory: An Introduction to Mathematics

5 Hensel’s Lemma 277

whereαN,αN+ 1 ,...,αN+n∈Sandan+ 1 ∈R.Since|an+ 1 πN+n+^1 |→0asn→∞,
the series

∑

n≥Nαnπ

nconverges with suma.

On the other hand, it is clear that ifa=

∑

n≥Nαnπ

n,whereαn∈SandαN=0,

then the coefficientsαnmust be determined in the above way.
IfFis complete then, by Corollary 11, any series

∑

n≥Nαnπ

nis convergent, since

|αnπn|→0asn→∞.

Corollary 14Every a∈Qpcan be uniquely expressed in the form

a=

∑

n∈Z

αnpn,

whereαn∈{ 0 , 1 ,...,p− 1 }andαn= 0 for at most finitely many n< 0. Conversely,
any such series is convergent with sum a∈Qp.Furthermorea∈Zpif and only if
αn= 0 for all n< 0.

Thus we have now arrived at Hensel’s starting-point. It is not difficult to show
that ifa=

∑

n∈Zαnp

n∈Qp, then actuallya∈Qif and only if the sequence of

coefficients(αn)iseventually periodic, i.e. there exist integersh>0andmsuch that
αn+h=αnfor alln≥m.
From Corollary 14 we can deduce again that the ringZof ordinary integers is
dense in the ringZpofp-adic integers. For, if

a=

∑

n≥ 0

αnpn∈Zp,

whereαn∈{ 0 , 1 ,...,p− 1 },then

ak=

∑k

n= 0

αnpn∈Z

and|a−ak|<p−k.

5 Hensel’sLemma

The analogy betweenp-adic absolute values and ordinary absolute values suggests
that methods well-known in analysis may be applied also to arithmetic problems. We
will illustrate this by showing how Newton’s method for finding the real or complex
roots of an equation can also be used to findp-adic roots. In fact the ultrametric
inequality makes it possible to establish a stronger convergence criterion than in the
classical case. The following proposition is modestly known as ‘Hensel’s lemma’.

Proposition 15Let F be a field with a complete non-archimedean absolute value||
and let R be its valuation ring. Let

f(x)=cnxn+cn− 1 xn−^1 +···+c 0