2
votes
1answer
32 views

p-adic numbers and GCD

Given two numbers $a,b \in \mathbb{Z}$, how do we prove that the $p$-adic number of $\gcd(a,b)$ is the same as the minimum for the $p$-adic number of $a$ and the $p$-adic number of $b$? Does this ...
6
votes
1answer
54 views

Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
0
votes
1answer
61 views

7-adic series expansion of square root of 2

Given the sequence $\{ a_n\}$ defined by the (positive and $a_n < 7^n$) solutions of the congruence $x^2 \equiv 2 \mod 7^n$ and $a_{n+1}\equiv a_n \mod 7^n$. e.g. the first one is $a_1 =3$ the ...
7
votes
1answer
116 views

Problems that are made easy by using p-adic numbers

Does anybody know of elementary problems that can be be solved using the p-adics? Solutions are preferred.
4
votes
3answers
338 views

Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$

Let G be the multiplicative group $\mathbb{Z}_n^*$ for $n = 2^k$ and $k \ge 3$. Can we prove that no element has order bigger than $2^{k-2}$ ? My solution (not really a solution) : Since $n=2^k$, I ...
-4
votes
1answer
147 views

Verifying that $\frac{p^n}{p^{n}-1}$ converges $p$-adically to $0$, while $\frac{1}{p^{n}-1}$ converges $p$-adically to $1$

This is a question from a book I'm struggling with, please could you provide a clear proof? Fix a prime number $p$. Verify that $\dfrac{p^n}{p^{n}-1}$ converges $p$-adically to $0$, while ...
2
votes
1answer
201 views

For what primes $p$ does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?

this is a question from a book I'm struggling with, please could you provide a clear proof For what primes p does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically? kind thanks
0
votes
1answer
63 views

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically and, when it does, to what limit?
0
votes
0answers
243 views

Show that the field of p-adic numbers is complete

this is a question from a book I'm struggling with, please could you provide a clear proof Show that the field of p-adic numbers is complete i.e. that a sequence of p-adic numbers converges if and ...
6
votes
1answer
185 views

Line in a proof on p69 in Cassel's Local Fields

I'm trying to read the proof of LEMMA 6.1 (Nagell) Let $u_n$ be defined by $u_0=0$, $u_1=1$ and $u_n=u_{n-1}-2u_{n-2} \hspace{20pt} (n\geq 2)$. Then $u_n=\pm1$ only for $n=1,2,3, ...
3
votes
1answer
83 views

$p$-adic closures of infinite sets

Let $S\subsetneq\mathbb{Z}$ be an infinite set. Does there always exist a prime $p$ such that the closure of $S$ in the $p$-adic integers, $\mathbb{Z}_p$, contains a rational integer $n\notin S$? ...
1
vote
2answers
112 views

Are there any non-trivial rational integers in the $p$-adic closure of $\{1,q,q^2,q^3,…\}$?

If $p$ is prime and not a divisor of $q$, are there any non-trivial rational integers in the $p$-adic closure of the set or powers of $q$? Edit: $q$ is also a (rational) integer, not a $p$-adic.