0
votes
1answer
64 views

Embedding into $p$-adic complex numbers

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind: Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the ...
7
votes
5answers
301 views

Proving $\sqrt{2}\in\mathbb{Q_7}$?

Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$? I understand Hensel's lemma, namely: Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers ...
0
votes
0answers
84 views

By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?

this is a question from a book I'm struggling with, please could you provide a clear proof Consider the sequence of rational numbers $a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1 ...
1
vote
1answer
197 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
229 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 ...
5
votes
1answer
121 views

When a number is a square in the p-adic rationals - proof question (Quadratic Residues)

I'm a little stuck with the proof of a theorem I'm trying to understand. The theorem is as follows: "For odd prime $p$, suppose for $\alpha \in Q_{p}$ (the p-adic rationals) that $|\alpha|_p=1$. Then ...
1
vote
0answers
92 views

Relationship between Connes trace formula and Weil's trace formula

Connes trace formula $$ \mathrm{Tr}\,{U(h)}=2h(1)\ln\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|} $$ Weil's trace $$ \int_{C}h(u)|u|d^{*}u- ...
4
votes
1answer
164 views

Is there a $p$-adic version of Liouville theorem?

That is, if a function $f$ is analytic and bounded in all $K$, a $p$-adic field (or more generally a complete non-archimedean field), has to be constant? And does the theorem work for functions on ...