The rings ,
Last update: 16 May 2012
The rings ,
Let be a prime (irreducible element of ).
- The -clock is the ring
where is the set of multiples of
in . Temporarily write for the
element in so that
we may distinguish from .
With this notation, the addition and multiplication in the ring
are given by
- The rings
and are the
subrings of given by
The real numbers contain the integers
and the addition and multiplication in are compatible with the addition and multiplication in .
The p-adic numbers contain the p-adic integers and the integers
and the addition and multiplication in and are compatible with the addition and multiplication in .
The rational functions
contain the formal power series
and the polynomials
where is the complex numbers and the addition and multiplication in and are compatible with the addition and multiplication in
Show that, in
Show that, in ,
The p-adic integers and the p-adic numbers
Let be prime.
The p-adic valuation on is
in its prime factorisation.
The fractional ideals of are the sets
The p-adic topology on is the topology generated by
as a system of fundamental neighborhoods of 0.
The p-adic numbers are the elements of , the completion of , wich respect to the p-adic topology, where the completion of a commutative topological group with fundamental system of neighborhoods of 0 is the completion with respect to the uniformity on generated by the sets
The p-adic integers are the elements of , the closure of in .
Show that is compact and open in .
Show that is a PID.
Show that is the unique prime ideal of and
be the projective system indexed by given by
Show that is the field of fractions of .
Show that is the completion of
in the -adic topology.
is another completion of . Why can't be written as the field of fractions of an inverse limit?
Notes and References
This section mostly follows [Bou, Top Gen III §6 Ex 23]. [Se, Ch.II], [AM, Ch.10], [Bou, Top Gen III §6 Ex 23-26] and [Bou, Top Gen III §7 Ex 1] are fundamental references. The book [Gou] provides a book length exposition.
M. Atiyah and I.G. Macdonald, Introduction to commutative algebra,
Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp.
General Topology, Springer-Verlag, 1989.
MR1488696 (98h:11155) F.Q. Gouvêa, p-adic numbers. An introduction Second edition, Universitext, Springer-Verlag, Berlin, 1997. vi+298 pp. ISBN: 3-540-62911-4
J.-P. Serre, A course in arithmetic, Translated from the French,
Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. viii+115 pp,