Number Theory: An Introduction to Mathematics

VI Hensel’sp-adic Numbers......................................

The ringZof all integers has a very similar algebraic structure to the ringC[z]of
all polynomials in one variable with complex coefficients. This similarity extends to
their fields of fractions: the fieldQof rational numbers and the fieldC(z)of rational
functions in one variable with complex coefficients. Hensel (1899) had the bold idea
of pushing this analogy even further. For anyζ∈C, the ringC[z] may be embedded in
the ringCζ[[z]] of all functionsf(z)=

∑

n≥ 0 αn(z−ζ)

nwith complex coefficientsαn

which are holomorphic atζ,andthefieldC(z)may be embedded in the fieldCζ((z))of
all functionsf(z)=

∑

n∈Zαn(z−ζ)

nwith complex coefficientsαnwhich are mero-

morphic atζ,i.e.αn=0 for at most finitely manyn<0. Hensel constructed, for each
primep,aringZpof all ‘p-adic integers’

∑

n≥ 0 αnp

n,whereαn∈{ 0 , 1 ,...,p− 1 },

and a fieldQpof all ‘p-adic numbers’

∑

n∈Zαnp

n,whereαn∈{ 0 , 1 ,...,p− 1 }

andαn=0 for at most finitely manyn<0. This led him to arithmetic analogues of
various analytic results and even to analytic methods of proving them. Hensel’s idea
of concentrating attention on one prime at a time has proved very fruitful for algebraic
number theory. Furthermore, his methods enable the theory of algebraic numbers and
the theory of algebraic functions of one variable to be developed completely in parallel.
Hensel simply definedp-adic integers by their power series expansions. We will
adopt a more general approach, due to K ̈ursch ́ak (1913), which is based on absolute
values.

1 ValuedFields...............................................

LetFbe an arbitrary field. Anabsolute valueonFis a map||:F→Rwith the
following properties:

(V1)| 0 |=0,|a|> 0 for all a∈F with a=0;
(V2)|ab|=|a||b|for all a,b∈F;
(V3)|a+b|≤|a|+|b|for all a,b∈F.

A field with an absolute value will be called simply avalued field.
Anon-archimedean absolute valueonFis a map||:F→Rwith the properties
(V1),(V2)and

(V3)′|a+b|≤max(|a|,|b|)for all a,b∈F.

W.A. Coppel, Number Theory: An Introduction to Mathematics, Universitext,
DOI: 10.1007/978-0-387-89486-7_6, © Springer Science + Business Media, LLC 2009

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