Completeness of the formal power series and valued fields

In summary, the conversation discusses a difficulty in showing a cauchy sequence in K[[t]] with a specific condition. The speaker mentions using a Cauchy sequence where the first nonzero term has a degree larger than n, and the limit of the sequence is a power series in K. The speaker also asks for clarification on how this approach can be applied to a similar problem.
  • #1
aalma
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1
TL;DR Summary
Trying to understand the completeness of K[[x]] and how to show that every cauchy sequence in it has a limit in K[[x]].
I had difficulty showing this no matter what I tried in (a) I am not getting it. Here for p(t) in K[[t]] : ## |p|=e^{-v(p)} ## where v(p) is the minimal index with a non-zero coiefficient.
I said that p_i is a cauchy sequence so, for every epsilon>0 there exists a natural N such that for all i,j>N we have
## |p_i(t)-p_j(t)|<epsilon ##, which is equivalent to that
## v(p_i(t)-p_j(t))>e^{-epsilon} ##.
But could not see how it helps here!.

Any clarifications would be great
 

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  • #2
In order for ##||p_i-p_j||## to be smaller than ##\varepsilon=e^{-n}## (for large ##i,j##) it must be the case that the first nonzero term of ##p_i-p_j## has degree larger than ##n##, i.e. the coefficients of ##1,t,...,t^n## are the same for ##p_i## and ##p_j.## So, for any fixed degree ##k##, the coefficient of ##t^k## in your sequence is eventually constant and the limit of your Cauchy sequence is just the power series whose coefficient of ##t^k## is this element of ##K##.
 
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Likes aalma
  • #3
Thanks!
Here is what I did in a.
can you give direction for b? I think it would be similar to a but could not see how..
20230129_222740.jpg
 

1. What does completeness mean in the context of formal power series and valued fields?

Completeness refers to the property of a mathematical structure to contain all possible elements that are necessary for a given operation or property to hold. In the context of formal power series and valued fields, completeness means that the structure contains all possible power series and field elements that are needed for certain operations or properties to be well-defined.

2. How is completeness related to convergence in formal power series and valued fields?

Completeness is closely related to convergence in formal power series and valued fields. A structure is considered complete if and only if every Cauchy sequence (a sequence in which the terms get arbitrarily close to each other) converges to a limit within the structure. In other words, completeness ensures that all possible convergent sequences are contained within the structure.

3. Can a formal power series or valued field be complete and still have elements that do not converge?

Yes, it is possible for a formal power series or valued field to be complete and still have elements that do not converge. This is because completeness only guarantees convergence for Cauchy sequences, not for all possible sequences. Therefore, there may still be elements that do not converge, but they are not necessary for the completeness of the structure.

4. How does completeness affect the algebraic properties of formal power series and valued fields?

Completeness has a significant impact on the algebraic properties of formal power series and valued fields. In a complete structure, certain algebraic operations, such as addition and multiplication, are well-defined and behave in the expected way. Additionally, completeness allows for the use of certain techniques, such as power series expansion and integration, which rely on the convergence of elements within the structure.

5. Are there any practical applications of completeness in formal power series and valued fields?

Yes, completeness has many practical applications in various fields of mathematics, including number theory, analysis, and algebra. For example, completeness is essential in the study of real and complex numbers, where it allows for the use of power series to approximate functions. It also plays a crucial role in the development of calculus and differential equations, where it allows for the use of integration and power series methods to solve problems.

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