Abelian Groups - László Fuchs [Springer]

18 1 Fundamentals
in Sect. 7 in Chapter 2). J p is again a ring, called the ring of p-adic integers, whose
non-zero ideals are p k J p with k D 0; 1; 2; : : : , and which is complete (i.e., every
Cauchy sequence is convergent) in the topology defined by its ideals.
For all practical reasons, the elements of J p may be represented as power series
in p. Note that f0; 1; : : : ; p 1g is a complete set of representatives of Z .p/ mod pZ .p/ ,
and more generally, f0; p k ;2p k ;:::;.p 1/p k g is a complete set of representatives of
p k Z .p/ mod p kC1 Z .p/ .Let 2 J p ,anda 0 ;:::;a n ;::: a sequence in Z .p/ converging
to (dropping to a subsequence, we may accelerate convergence, and assume
a n 2 p n J p for all n 2 N). Owing to the definition of convergence, almost
all a n belong to the same coset mod pZ .p/ , say, to the one represented by some
s 0 2f0;1;:::;p 1g. Almost all differences a n s 0 belong to the same coset of
pZ .p/ mod p 2 Z .p/ , say, to the one represented by s 1 p. So proceeding, defines a
sequence s 0 ; s 1 p;:::;s n p n ;:::, which is the same for every sequence converging
to . Accordingly, we assign to the (formal) infinite series
D s 0 C s 1 p C s 2 p 2 CCs n p n C ::: .s n 2f0;1;:::;p 1g/:
Its partial sums b n D s 0 C s 1 p C C s n p n .n D 0;1;2;:::/ form a Cauchy
sequence in Z .p/ that converges to in J p because b n 2 p k J p for n k.Fromthe
uniqueness of limits it follows that in this way different elements of J p are associated
with different series, and since every infinite series s 0 Cs 1 pCCs n p n C::: defines
an element 2 J p , we may identify the elements of J p with the corresponding series.
Let us see how to compute in J p .If D r 0 C r 1 p C C r n p n C ::: with
r n 2f0;1;:::;p 1g is another p-adic integer, then the sum C D t 0 C t 1 p C
Ct n p n C ::: and the product D v 0 C v 1 p CCv n p n C ::: are computed as
follows: t 0 D r 0 C s 0 `0p; t n D r n C s n C `n 1 `np and v 0 D r 0 s 0 m 0 p;v n D
r 0 s n Cr 1 s n 1 CCr n s 0 Cm n 1 m n p for n D 1;2;::: where the integers `n; m n are
uniquely determined by the rule that all of t n ;v n are integers in the set f0;:::;p 1g.
As to subtraction and division, note that if, e.g., s 0 ¤ 0, then the negative of is
D .p s 0 /p C .p s 1 1/p C .p s 2 1/p 2 C :::,andtheinverse 1 of
exists if and only if s 0 ¤ 0 in which case it may be computed by using the inverse
rule of multiplication.
We shall denote by J p both the ring and the group of the p-adic integers. Q p will
denote the field of quotients of J p (and its additive group); its elements are of the
form p k with 2 J p ; k 2 Z.
F Notes. It will perhaps be instructive to mention a few important applications of abelian
groups, in particular, to illustrate the groups in this section by pointing out a few applications
outside group theory.
The most widespread applications of abelian groups outside algebra are in algebraic topology
and algebraic geometry. A main step was the reinterpretation of Betti numbers as invariants of
finite groups. In algebra, the additive groups of rings and the unit groups of commutative rings
play a leading role, see Chapter 18.Elementaryp-groups are the additive groups of fields of prime
characteristics. The case p D 2 plays a prominent role in computer theory as well as in coding
theory. Class groups of integral domains are abelian groups. As a matter of fact, L. Claborn [Pac.
J. Math. 18, 219–222 (1966)] proved that every abelian group occurs in this way.