In mathematics the $p$-adic number system for any prime number $p$ extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems.

learn more… | top users | synonyms

2
votes
0answers
22 views

Convergence with the $5$-adic metric.

I'm struggling to get any intuition for the following example: Consider the sequence $\{a_k\} = \{3, \ 33, \ 333, \ 3333, \ \ldots\}$ It's easy to show, using the geometric series formula, that $a_k ...
2
votes
1answer
30 views

Index for p-adic subgroups

I always got some problems in computing precisely and understanding indexes for congruence-like subgroups. My problem seems quite simple: what is the index of $(1 + p^r)^2$ in $\mathbf{Z}_p^\times$ (I ...
3
votes
1answer
40 views

Valuation of Index of polynomial with Newton Polygon

I read here (page 237) that the valuation of the index of a polynomial is equal to the number of integer points below its Newton polygon. I am confused how this makes sense--the cited paper (this) ...
3
votes
0answers
63 views
+100

Fraction field of $p$-adic power series ring

Let $L$ be a finite extension of $\mathbf{Q}_p$. Write $$\mathcal{O}_{\mathcal{E}} = \left\{ f = \sum_{k \in \mathbf{Z}} a_kT^k \in \mathcal{O}_{L}[[T,T^{-1}]] \mid \lim_{k \to -\infty} a_k = 0\right\...
0
votes
0answers
30 views

What is the $2$-adic order of the number $(2k+1)^n-1?$

If is possible to calculate ${\rm ord}_2((2k+1)^n-1)$ where ${\rm ord}_2(N)$ means the greatest non-negative integer $\nu$ such that $2^\nu$ divides $N$ and $n$ is odd? After simplification I have ...
1
vote
2answers
126 views

An example of a discontinuous “$\ell$-adic Galois representation”

Let $\mathbb{F}_p$ be a finite filed with $p$ elements, and $G=\mathop{\mathrm{Gal}(\mathbb{F}_p^s/\mathbb{F}_p)}$ be its absolute Galois group. $G$ is a pro-finite group, with the Krull topology, see ...
1
vote
1answer
61 views

2-adic valuation of odd harmonic sums

I'm playing with p-adic valuations, and find that the odd harmonic sums, $\tilde{H}_k=\sum_{i=1}^{k}\frac{1}{2i-1}$, has 2-adic valuation $||k^2||_2=2||k||_2$. E.g.) $\tilde{H}_4=\frac{176}{85}$ has ...
0
votes
1answer
35 views

p-adic valuation

My question is two-fold. First off, let's define $v_p$ to be the valuation on $\mathbb{Q}$ defines by setting $v_p( \frac{a}{b} p^n)=n$, where $(a,p)=(b,p)=1$. How exactly does $v_p$ extend to $\...
0
votes
0answers
20 views

zero of $p$-adic L-functions

Does these two $p$-adic $L$-functions has zero in $\mathbf{Z}_p$ : $L_p(s, \omega^{p-i})$ when $(p, i)=(43867, 17)$ and $(657931, 25)$? I know that the if $p$ divides the Bernoulli number $B_{i+1}$ if ...
2
votes
2answers
43 views

p-adic distances

We take $\mathbb{Q}_p$ to be the completion of $\mathbb{Q}$ with respect to $|\cdot|_p$. If $x=\sum_{j=k}^{\infty} a_jp^j$ is some element in $\mathbb{Q}_p$, then how exactly does $|\cdot|_p$ extend? ...
3
votes
1answer
38 views

Analytic continuation on $\mathbb C_p$

Let $p$ a prime number and $f(z)=\prod_{n\ge0}\left(1-z^{p^n}\right)$. One sees easily that $f$ is defined on $\left\{z\in\mathbb C_p\mid v_p(z)>0\right\}$. But can one continue it analytically on ...
5
votes
2answers
128 views

Polynomials generating the same $p$-adic fields

I wonder if the following fact is true: Pick $l\in \mathbb N$ a number and let $f,g\in \mathbb Z_p[x]$ be monic polynomials with coefficients in the ring of $p$-adic integers such that $f\equiv g \...
2
votes
0answers
36 views

What is the algebraic structure of $\Bbb Q_p/\Bbb Z_p$? [closed]

I am curious about the algebraic structure of $\Bbb Q_p/\Bbb Z_p$. Is there any result in this direction? Thanks!
1
vote
2answers
41 views

An equation is solvable in $\mathbb Z_p$

Suppose $p$ is an odd prime and $a\in \mathbb Z_p$ with $a\equiv 1 \mod p^2 $. Then $x^p=a$ is solvable over $\mathbb Z_p$. I want to prove $x^p\equiv a\mod p^v$ is solvable for arbitrary $v\geqslant ...
0
votes
1answer
27 views

$\mathbb{Z}_p$-extensions of CM-fields

I am trying to prove some consequences of Iwasawa's Theorem for CM-fields. There is a sequence of CM-fields $$K=K_0\subseteq K_1 \subseteq \dots \subseteq K_\infty$$ so that $K_\infty/K$ is a $\mathbb{...
1
vote
1answer
39 views

Deducing Skolem-Mahler-Lech theorem from a $p$-adic interpolation result.

Let $p$ be a prime number, let $d$ be an integer greater than $1$ and let $f\in\mathbb{Z}_p[X_1,\cdots,X_d]^d$. I have already proved the following $p$-adic interpolation theorem: Theorem 1. (...
13
votes
2answers
1k views

Galois group over $p$-adic numbers

Can one describe explicitly the Galois group $G=\operatorname{Gal}(\overline{\mathbb Q_p}/\mathbb Q_p)$? I only know the most basic stuff: unramified extensions of $\mathbb Q_p$ are equivalent to ...
18
votes
0answers
247 views

Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units

On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem: Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\...
1
vote
1answer
50 views

$\mathbb{Z}_p$ is an Integral Domain

Assume that we define the ring of the $p$-adic integers as the projective limit $$\mathbb{Z}_p =\varprojlim \frac{\mathbb{Z}}{p^n\mathbb{Z}}$$ Then $\mathbb{Q}_p$, the field of the $p$-adic numbers ...
0
votes
1answer
35 views

Codomain of p-adic logarithm

We have the natural map $$\log: \mathbb{C}^\times \to \mathbb{R}$$ $$z \to \log |z|$$ Is there a p-adic analogue of this? By this I mean, a map $\log_p: \mathbb{C}_p^\times \to \mathbb{Q}_p$, ...
9
votes
0answers
90 views

Integer polynomials with roots in every $\mathbb{Z}_p$ but no rational roots.

I want to find polynomials in $\mathbb{Z}[x]$ with degree as small as possible such that these polynomials have no rational roots but have a root in the $p$-adic integers $\mathbb{Z}_p$ for every ...
0
votes
0answers
23 views

Embeddings of $K_v$ in $\mathbb{C}$

Let $K$ be a number field, $v$ a nonarchimedean prime, and $K_v$ the completion of $K$ at $v$. We have the embedding $K \to K_v$, and also $K \to \mathbb{C}$. I have two related questions: Is ...
2
votes
1answer
54 views

Isomorphism which involves $\mathbb Z_p[[T]] \otimes \mathbb Q_p$

Why should $\mathbb Z_p[[T]] \otimes_{\mathbb Z_p} \mathbb Q_p$ be isomorphic to the bounded sequences with values in $\mathbb Q_p$? The fact is that the tensor product is on $\mathbb Z_p$, so it is ...
1
vote
0answers
22 views

Metric on the profinite completion of the integers?

The p-adic integers come with a metric and associated topology, both of which can be restricted down to the integers. Does this also apply to the profinite completion of the integers? Do they have ...
1
vote
2answers
60 views

Restriction from subgroup of the Galois group of max. unr, ext. $G(\tilde{K}/\mathbb{Q}_{p})$ to $G(K/\mathbb{Q}_{p})$ is surjective?

This is a question I'm struggling with for some time. Let $K$ be a finite Galois extension of $\mathbb{Q}_{p}$ and let $\tilde{K}$ denote the maximal unramified extension of $K$. We can then ...
2
votes
0answers
46 views

Formal power series over p-adic integers

Does $\mathbb Z_p \otimes \mathbb Z[[x]]=\mathbb Z_p[[x]]$ hold? In particular, I don't know how to express $\sum\frac 1{q^n}x^n$ for $q\ne p$ as an element of the tensor product. Or it should be $\...
2
votes
0answers
40 views

Introduction to p-adic vector spaces

I'm interested in learning about vector spaces over $\mathbb{C}_p$ and $\mathbb{Q}_p$. Most textbooks on p-adic numbers (Koblitz, Schikhof) focus on analysis and number theory. Is there any ...
4
votes
1answer
46 views

Upper Numbering of Ramification Groups of Absolute Galois Groups for Totally Ramified Extensions

Suppose $K'/K$ is a totally ramified extension of $p$-adic fields of degree $e.$ A paper (p.9, line 15) I am reading seems to use the following formula for the upper numbering on the absolute galois ...
6
votes
2answers
74 views

A number $a$ is a square in $\mathbf{Q}$ if and only if it is a square in $\mathbf{R}$ and $\mathbf{Q}_p$ for all primes $p$

Problem from Schikhof's Ultrametric Calculus. As I understand it, the intersection of $\mathbf{R}$ and all $\mathbf{Q}_p$ is just $\mathbf{Q},$ so it seems that $x^2-a$ having a zero in $\mathbf{Q}$ ...
2
votes
1answer
50 views

zeroes of homogeneous analytic $p$-adic functions

I am trying to understand Lemme 2.1 page 3 of this paper by Pilloni. What is says (I think) is that if you have, for a a positive real number $w$, an analytic function $$ f : \mathbf{Z}_p^\times(1+...
1
vote
0answers
28 views

$p$-adic Fourier transforms and orthogonality relations

In $\mathbb{C}$, we have the following orthogonality relation $$ \int_{0}^{1} e^{2\pi i (m-n)x} dx = \begin{cases} 1 & \mbox{ if } m = n;\\ 0 & \mbox{ otherwise.} \end{cases} $$ Do we have ...
1
vote
0answers
24 views

Solutions in $\mathbb Q_p$ leads to solution for congruences equations?

Let $p$ be a prime number such that $p\equiv1\pmod 3$. Let $n$ be an integer such that the equation $x^3=n$ has a solution in $\mathbb Q_p$. In fact with our assumptions, the others solution are in $\...
2
votes
2answers
50 views

Axiomatic Approach to p-adic rationals

There are several ways to construct the real numbers: completion of the rationals w.r.t. the euclidean metric, dedekind completion of the rationals, infinite continued fractions, etc. Each of these ...
1
vote
0answers
49 views

Limit of a sequence in $\mathbb{Z}_p$ (J.-P. Serre, p-adic equations)

In a proof of a theorem in chapter 2 "p-adic equations" in "A Course in Arithmetic" from J-P Serre there is one conclusion that I don't understand. Here is the theorem I'm talking about (excluding the ...
1
vote
1answer
44 views

Galois group action on etale cohomology groups

Let $X$ be a smooth and proper scheme over $Spec(\mathbb{Z}_p)$. Let $l$ be a prime number coprime to $p$. Then the proper base change theorem gives me an isomorphism $$H^r_{et}(X\times_{\mathbb{Z}_p}\...
115
votes
14answers
15k views

Why does an argument similiar to 0.999…=1 show 999…=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the ...
0
votes
0answers
41 views

Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
2
votes
0answers
53 views

Embedding a number field in $\mathbb{Q}_p$.

Let $K/\mathbb{Q}$ be a finitely generated field extension and let $(x_1,\cdots,x_m)$ be a transcendance basis of $K/\mathbb{Q}$. Using primitive element theorem, there exists $y\in K$ algebraic over $...
0
votes
0answers
16 views

A degree comparison problem over p-adic fields

Consider multivariate polynomial $p(x)\in\Bbb Q_p(x)$ and multivariate rational function $r(x)\in\Bbb Q_p(x)$ where $x=(x_1,\dots,x_n)$. Assume that degree in each variable is $1$ but total degree ...
0
votes
1answer
39 views

Isometries of p-adic vector spaces?

In Euclidean space $\mathbb{R}^d$ there are ${d+1 \choose 2}$ independent isometries (translations and rotations). In other words the dimension of the Euclidean isometry group is ${d+1 \choose 2}$. ...
0
votes
1answer
46 views

Understanding a proof of a corollary in chapter 2 about invertibility of a p-adic integer (Jean-Pierre Serre)

In a proof of a corollary in chapter 2, there is a step I don't understand. Corollary 2: Suppose $p \neq 2$. Let $f(X) = \sum_j a_{ij}X_iX_j$ with $a_{ij} = a_{ji}$ be a quadratic form with ...
0
votes
1answer
41 views

A question about the definition of $p$-adic pseudo-measure.

Let $\mathfrak B$ be a profinite abelian group and let $\Lambda(\mathfrak B)$ be defined as the inverse limit $\varprojlim \mathbb Z_p[\mathfrak B/ \mathcal H]$ where the inverse limit is taken with ...
1
vote
1answer
23 views

P-adic expansion of rational

I want to find the 7adic expansion of 1/4. I found that this is …1515152, by using the algorithm of finding 1/4 = k + 7q for each digit. Is this correct?
1
vote
0answers
17 views

Changing uniformizer of $p$-adics

In the theory of $p$-adic fields typically a uniformizer $\pi$ is chosen that generates the maximal ideal, $m$. And a few theorems later it can be shown that every element $x \in O$ of the ring of ...
2
votes
0answers
33 views

Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
1
vote
0answers
49 views

II. p-adic equations [2.1. Solutions] (J.-P. Serre)

Currently, I'm reading a chapter about p-adic equations in A Course in Arithmetic by Jean-Pierre Serre and I have a hard time understanding it. My questions/thoughts are in textboxes like these. ...
3
votes
1answer
65 views

Subgroups of finite index of the split maximal torus of $GL_n(\mathbb{Z}_p)$.

Let $E_{i,j}$ be the $n \times n$ elementary matrices. Let $G=GL_n(\mathbb{Z}_p)$. Let $T_G$ be the split maximal torus of $GL_n(\mathbb{Z}_p)$. Let $\Theta$ be the subgroup of $T_G$ consisting of ...
2
votes
1answer
34 views

Limit of p-adic numbers

Let $\alpha\in\mathbb{Z}_p^\times$. I read somewhere that the limit of $\alpha^{n!}$ as $n\rightarrow\infty$ is equal to one. Can someone explain to me why this is? Thanks!
5
votes
0answers
45 views

Proving that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers)

I want to prove that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers). By the definition of inverse limit, we know that there is a ring homomorphism $\phi$ from $\mathbb Z_3$ to $\mathbb Z/...
4
votes
2answers
41 views

Rational sum of the $p$-adic series

Koblitz states (as one of the excercises to chapter 2) that whenever we are given an integer $k > 0$ and prime $p$, the series $$ f(p, k) = \sum_{n=0}^\infty n^kp^n $$ converges in $\mathbb Q_p$ ...