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Completions of fields
*September 1, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.*

Tags: absolute values, Cauchy sequences, completions, p-adic numbers

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Tags: absolute values, Cauchy sequences, completions, p-adic numbers

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So again, we’re back to completions, though we’re going to go through it quickly. Except this time we have a field with an absolute value like the rationals with the usual absolute value.

**Completions **

Definition 1The(more…)completionof is defined as the set of equivalence classes of Cauchy sequences:

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Generalities on completions
*August 23, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra, topology.*

Tags: Cauchy sequences, completions, inverse limits

3 comments

Tags: Cauchy sequences, completions, inverse limits

3 comments

Today I’ll discuss completions in their algebraic context. All this is really a version of Cauchy’s construction of the real numbers, but it’s also useful in algebra, since one can study a ring through its completions (e.g. in algebraic number theory, as I hope to get to soon).

**Generalities on Completions **

Suppose we have a filtered abelian group with a descending filtration of subgroups . Because of this, we can consider “Cauchy sequences” and “convergence:”

Definition 1The sequence , isCauchyif for each , there exists large enough that

The sequence

converges toif for each , there exists large enough that