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, , and the ring of Witt vectors

No.06:

DEFINITION 06.1   We denote by the quotient field of .

LEMMA 06.2   Every non zero element is uniquely expressed as

We have so far constructed a ring and a field for each prime .

PROPOSITION 06.3   Let be a prime. Then:
1. is a local ring with the unique maximal ideal .
2. is an integral domain whose quotient field is a field of characteristic zero.

With and/or , we may do some calculus'' such as:

THEOREM 06.4   [1, corollary 1 of theorem 1] Let , . Assume that there exists a natural number such that ,

Then there exists such that

 (1) (2)

See [1] for details.

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2008-06-10