Articles of p adic number theory

$p^a\mid f(v) \implies p^a\mid f(w)$ in $\mathbb Z$

Let $p$ be a prime, $n$ a positive integer and $q=p^n$ a $p$- power. Let $w,v$ is a root of the $q,p$-th cyclotomic polynomial $Q_q,Q_p$, respectively. Let $f$ be a polynomial with coefficient in $\mathbb Z$. Prove or disprove: “$p^a\mid f(v) \implies p^a\mid f(w)$” in $\mathbb Z[w]$. Edit: This question was an “iff” question, now […]

Solve $f^n(x)=4^nx+\frac{4^n-1}{3}$ for $n$.

Solve $y=4^nx+\frac{4^n-1}{3}$ for $n$, where $n\in\mathbb{N_{\geq0}},$ $y\in2\mathbb{N}_{\geq0}+1$ and $x\in2\mathbb{N}_{\geq0}+1\setminus(4\mathbb{N}_{\geq0}+1\setminus8\mathbb{N}_{\geq0}+1)$. In clearer language the process is to start with some odd integer and subtract one and divide by $4$ repeatedly until you hit some number which would take you out of the odd integers if you continued further. The question asks you to find for any […]

Valuations, Isomorphism, Local ring

Let $x \in \mathbb{Q}, \, x \neq 0$, such that $x=p^r \frac{a}{b}, \quad a,b, r \in \mathbb{Z}, \quad p \nmid a, \quad p\nmid b$. Let $v_p(x):=r$ and $v_p(0):= \infty$. Also, $$\mathcal O_p= \left\{ x \in \mathbb{Q} : v_p(x) \geq 0\right\}= \left\{\frac{a}{b} : p \nmid b\right\} \quad \& \quad m_p= \left\{ x \in \mathbb{Q} : v_p(x) […]

Computing the quotient $\mathbb{Q}_p/(x^2 + 1)$

This question already has an answer here: Decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields 1 answer

Applications of the p-adics

As I was preparing to spend a month studying p-adic analysis, I realized that I’ve never seen the theory of p-adic numbers applied in other branches of mathematics. I can certainly see that the field $\mathbb Q_p$ has many nice properties, e.g. $X\subset Q_p$ closed and bounded implies $X$ compact, $\sum\limits_{n=0}^\infty x_n$ exists iff $\lim\limits_{n\to\infty}x_n=0$, […]

On a $p$-adic unit and the existence of its $n$-th root

Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\alpha$ be a $p$-adic unit, i.e. an invertible element of the multiplicative monoid $\mathbb{Z}_p$. Consider the set $S = \{n \in \mathbb{Z}, n \gt 0\mid x^n = \alpha$ has a solution in $\mathbb{Q}_p\}$. Is $S$ an infinite set?

How do you take the multiplicative inverse of a p-adic number?

I am reading the wiki page for p-adic numbers and it states that they are a field extension of the rationals so each member has to have a modular multiplicative inverse. So how would I take the inverse of, say, 35 in the ring of 2-adic numbers?

Profinite and p-adic interpolation of Fibonacci numbers

On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here) For each profinite integer $s$, one can in a natural way define the $s$th Fibonacci number $F_s$, which is itself a profinite integer. Namely, given $s$, one can choose a sequence of positive integers $n_1, n_2, n_3,\dots$ that […]

p-adic liftings on SAGE

I asked a question the other day: Multidimensional Hensel lifting which @Hurkyl kindly and very elegantly answered. A follow-on from this is that I have tried to implement exactly the “algorithm” implicit in his answer in order to solve my problem on SAGE, but without much success. Has anyone out there had any success with […]

Roots in different algebraic closure have the same multiplicative relations

Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, with $f(0) \ne 0$, and $p$ a prime number. Also, let $a_1, \ldots, a_n \in \overline{\mathbb{Q}}_p$ be all the roots of $f$ taken in the algebraic closure of the $p$-adic numbers, and let $b_1, \ldots, b_n \in \mathbb{C}$ be all the roots of $f$ taken in […]