power series and strings for p-adics

[Fernando Gouvea]

If I create the p-adics in the default way, p-adic numbers are power series:

K=Qp(7)
a=K(1/42)
print a

6*7^-1 + 5 + 5*7 + 5*7^2 + 5*7^3 + 5*7^4 + 5*7^5 + 5*7^6 + 5*7^7 + 5*7^8 + 5*7^9 + 5*7^10 + 5*7^11 + 5*7^12 + 5*7^13 + 5*7^14 + 5*7^15 + 5*7^16 + 5*7^17 + 5*7^18 + O(7^19)

On the other hand, I can ask for p-adic digits too:

Kd=Qp(7,print_mode="digits")
a=Kd(1/42)
print a

...5555555555555555555.6

In certain contexts, I'd like to work in the power series mode up until the end, but display my answer in digits mode. The only way I know how to do this is to find the answer w and then use answer=Kd(w) to turn it into digits. Is there an easier way? I expected something like print a.digits() to work, but there's no such attribute.

Thanks,

Fernando

-- 
=============================================================
Fernando Q. Gouvea         http://www.colby.edu/~fqgouvea
Carter Professor of Mathematics
Dept. of Mathematics and Statistics
Colby College              
5836 Mayflower Hill        
Waterville, ME 04901       

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[Nils Bruin]

On Thursday, August 8, 2019 at 7:16:21 AM UTC-7, Fernando Gouvea wrote:

In certain contexts, I'd like to work in the power series mode up until the end, but display my answer in digits mode. The only way I know how to do this is to find the answer w and then use answer=Kd(w) to turn it into digits. Is there an easier way? I expected something like print a.digits() to work, but there's no such attribute.

The print-mode does not affect how the computations are done. So your work will be done the same way no matter what print-mode is selected

To access the p-adic digits, you can use "Qp(7)(1/42).list()"

You can give a mode to the "str" method that returns a string representation. e.g. Qp(7)(1/42).str(mode="bars") gives a presentation that makes the digits obvious. In principle Qp(7)(1/42).str(mode="digits") should work but it seems to miss some initialization. You could work around that by coercing into another p-adic ring with the desired print options set:

k1=Qp(7)

k2=Qp(7,print_mode="digits")

a=k1(2).sqrt()

print k2(a)

[Dominique Laurain]

Hello Fernando,

I want just to say that I liked a lot your book "Introduction to p-adic numbers" when I bought and read it some years ago.

A very good fun "travel" in number theory. I appreciate the way it process by extending fields.

I tried to code something in Sage too (Hensel Lemma) but I am no clearly satisfied about it : ...some tendency by sagemath...to convert to "fixed field" (rational)..but too much difficult to explain by me.

Dominique.