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
add a comment
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 1 The completion of is defined as the set of equivalence classes of Cauchy sequences: (more…)
Generalities on completions August 23, 2009
Posted by Akhil Mathew in algebra, commutative algebra, topology.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 1
The sequence , is Cauchy if for each , there exists large enough that
The sequence converges to if for each , there exists large enough that