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

0
votes
1answer
39 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
35 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
17 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?
0
votes
0answers
15 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
27 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 ...
-2
votes
0answers
28 views
1
vote
0answers
41 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
57 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
30 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
37 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 ...
3
votes
2answers
29 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 $\mathbb Q_p$ ...
2
votes
0answers
29 views

Stronger form of Hensel's lemma?

Let $f \in \mathbb{Z}_p[x]$ and suppose $|f(a)|_p < |f'(a)|_p^2$ for some $a \in \mathbb{Z}_p$. Let $a_1 = a$, and for $n \ge 1$ let$$a_{n+1} = a_n - f(a_n)/f'(a_n).$$How do I see that this defines ...
5
votes
1answer
72 views

If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$?

If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$? I know we want to use Hensel's Lemma somehow to assess this question, but I'm ...
0
votes
0answers
23 views

Constructing Witt polynomials

I am reading about Witt vectors, and I keep seeing the following set of congruences often: For example, in these notes here, on page 3 we see the following congruences: \begin{align} ...
2
votes
2answers
38 views

Please express the first 3 7-adic digits of a root of $x^3-1$ in $\mathbb{Z}_7$ other than 1.

Please express the first 3 p-adic digits of a root of $x^3-1$ in $\mathbb{Z}_7$ other than 1. Does this just mean find the 7-adic expansion of 2 or 4? Wouldn't their expansions just be 2 and 4?
0
votes
0answers
27 views

P-adic expansions of Integers

Is there a way to prove that $a$ is an element of the Integers, if $p$ is prime, $a$ is in $\mathbb Z_p$ and its $p$-adic expansion is eventually periodic? I know how to do this for the ...
7
votes
2answers
138 views

$p$-adic valuation of harmonic numbers

For an integer $m$ let $\nu_p(m)$ be its $p$-valuation i.e. the greatest non-negative integer such that $p^{\nu_p(m)}$ divides $m$. Let now $H_n=1+\dfrac{1}{2}+ \cdots+ \dfrac{1}{n}$. If ...
0
votes
0answers
35 views

$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
0
votes
1answer
45 views

Majoration of the $p$-adic valuation of a factorial.

Let $p$ be a prime number. In order to prove a result on $p$-adic interpolation of iterates, I need to show the following: Lemma. Let $m$ be an integer, one has: ...
11
votes
2answers
61 views

Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $

I read in a paper that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $ this is counterintuitive / surprising since $\mathrm{SO}_3(\mathbb{R}) \not \simeq \mathrm{SL}_2(\mathbb{R}) $ ...
2
votes
0answers
95 views

What is a “branch” of p-adic exponentiation?

I am reading p-adic Numbers, p-adic Analysis, and Zeta-Functions by Neal Koblitz. Please look at page 27. The definition of $f(x)=n^x$ is unambiguously defined when $n$ is $1 \mod p$. Next, for any ...
1
vote
1answer
38 views

Can the zeroes of a multivariate $p$-adic polynomial be bounded?

Multivariate real polynomials, as opposed to multivariate complex polynomials, can have bounded zero sets, i.e. $x^2+y^2-1$ in $\mathbb{R}^2$. This fails in $\mathbb{C}^n$ because $\mathbb{C}$ is ...
4
votes
1answer
55 views

Find a matrix of order $p$ in a subgroup of $\operatorname{GL}_n(\mathbb Z_p)$

Let $p$ be a fixed prime and $\mathbb Z_p$ the ring of $p$-adic integers. Consider the subgroup $G_n\subseteq \operatorname{GL}_n(\mathbb Z_p)$ given by all matrices $(a_{ij})_{ij}$ such that $$ ...
4
votes
1answer
44 views

Do the number of degree p extensions of p-adic fields lie in a recursive sequence? And if so, why?

I noticed something on this page, that may just be coincidental: http://www.lmfdb.org/LocalNumberField/ From inspecting the table there, you can conclude that most of the interesting extensions of ...
1
vote
0answers
35 views

Serre mass formula under extension

Suppose $K_1$ is a local field and $K_2$ is a totally ramified finite Galois extension. Let $e$ be a positive integer with $[K_2:K_1]|e$. Consider the set of isomorphism classes of $K_1$ extensions of ...
5
votes
1answer
962 views

What are the quadratic extensions of $\mathbb{Q}_2$?

How do you classify the non-squares in $\mathbb{Q}_2$? I've tried writing down expansions for "odd" numbers in $\mathbb{Z}_2$, but unlike in $\mathbb{Z}_p$, the n$^{th}$ term in the expansion is not ...
113
votes
13answers
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 ...
3
votes
0answers
53 views

“Logarithmic derivative of a p-adic number”

Coleman, in "Division Values in Local Fields", (Inventiones math. 53, 91 - 116 (1979)), says that In his work on cyclotomic fields Kummer observed that various formal operations on power series ...
2
votes
0answers
69 views

infinitely $p$-divisible elements in $A\otimes \mathbb{Z}_p$

Let $A$ be a (possibly non-finitely generated) torsion-free abelian group. Suppose that $A$ contains no infinitely $p$-divisible elements, then does the same hold for $A\otimes \mathbb Z_p$, where ...
2
votes
1answer
37 views

Where do these p-adic identities come from?

I was reading this article (http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf) to see some applications of $p$-adic numbers outside mathematics, and came across these two identities: ...
1
vote
1answer
22 views

A badly convergent p-adic series from one of the Schikhof's excercises

W. H. Schikhof in his book on Ultrametric calculus suggests to solve the following problem: Exercise 23.J (van Hamme) Use the ideas of the previous excercise to show that in $\mathbb Q_p$ ($p \neq ...
8
votes
0answers
40 views

Is torsion of a topological module closed?

I was asking to myself the following question. Consider a $p$-adically complete and separated topological algebra $R$ over $\mathbb{Z}_{p}$. As $\mathbb{Z}_{p}$ is a domain, we know that the ...
5
votes
3answers
152 views

Why does taking completions make number fields simpler?

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example, If $K$ is a finite extension of $\mathbb ...
4
votes
1answer
392 views

What is the index of the $p$-th power of $\mathbb Q_p^\times$ in $\mathbb Q_p^\times$

In this book it is listed as an exercise to compute the index $[\mathbb Q_p^\times:(\mathbb Q_p^\times)^p]$. This exercise is appended to a section concerning the structure of unit-group filters, ...
1
vote
1answer
49 views

What does extra zero of an $L$-function mean?

This is a very vague question. What does an extra zero of an $L$-function mean? There are lots of papers written on this topic, investigating the extra/exceptional zeros of various $p$-adic ...
1
vote
2answers
35 views

Finite extensions of $\mathbb Q_p$ are exactly completions of numberfields

I read that every finite extension of $\mathbb{Q}_p$ is in fact a completion of a numberfield K with a place of K. I also heard that this is a consequence of Krasner´s Lemma. Do you have any hint how ...
3
votes
1answer
67 views

Completions of number fields

I would like to prove a statement about completions of number fields, but I'm running into a problem. The statement I want to prove is Let $L/K$ be a Galois extension of number fields, $p$ a prime ...
2
votes
1answer
41 views

Why must a zero of $f \in \mathbb{Z}_p[X_1, \dots, X_m] ~ (\text{mod } p^n)$ be simple in order to lift to $\mathbb{Z}_p$?

In chapter II, section 2.2, of J-P. Serre's A Course in Arithmetic, we have the following theorem: Theorem 1: Let $f \in \mathbb{Z}_p[X_1, \dots, X_m]$, $x = (x_i) \in \left( \mathbb{Z}_p ...
0
votes
0answers
28 views

How to show that some equation does not have any solutions in $\mathbb Q_p^2$.

Is it true that if an equation has solutions in $\mathbb Q^2$, it has a solution in $\mathbb Q_p^2$ for all primes $p$? For example, if $f(a, b) = a^2 - 2b^2$, the only solution of $f(a,b) = 0$ in ...
2
votes
0answers
64 views

What constitutes a good reading course in $p$-adic number theory?

I have had a course in number theory where I studied Marcus and also a course in differential geometry. I have read Koblitz's introductory book on $p$-adic numbers. I am roughly interested in both ...
1
vote
1answer
55 views

Congruence subgroup of $\mathbb{GL}_n(\mathbb{Z}_p)$

In course of my research I met the following situation : 1) I have a bunch of open subgroup (so of finite index) in $\mathbb{GL}_{n}(\mathbb{Z}_p)$. 2) My groups arises naturally as stabilizers of ...
1
vote
1answer
26 views

Find the first six terms in a reduced coherent sequence defining $i$ in $\mathbb{Z}_5$.

Find the first six terms in a reduced coherent sequence defining $i$ in $\mathbb{Z}_5$. I need to use the sequence $a_k=2^{5^{k-1}}$ but not sure how to? Any hints?
0
votes
0answers
38 views

$4$th root of unity: 5-adic

For $k \geq 1$, let $x_k = a^{p^{k-1}}$. Taking $p = 5$ and $a = 2$, find the first six terms in a reduced coherent sequence defining a $4$th root of unity (i.e. $\sqrt{−1}$) in $\mathbb{Z_5}$, and ...
0
votes
2answers
229 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 ...
4
votes
2answers
72 views

${\mathbb{Q}_p^*}^2$ is open in $\mathbb{Q}_p^*$

Show that the set of squares in $\mathbb{Q}_p^*$ is open in $\mathbb{Q}_p^*$. Here $\mathbb{Q}_p$ is the $p$-adic numbers and $\mathbb{Q}_p^*$ is the set of units in $\mathbb{Q}_p$. I know that ...
1
vote
1answer
53 views

$\mathrm{ord}_p(x)$ and convergence in $\mathbb{Q}_p$

Let $x=\frac{22}{7} \in \mathbb{Q}$. (a) Find $\mathrm{ord}_p(x)$ for all primes $p$. $\mathrm{ord}_2(x)=1,\ \mathrm{ord}_{11}(x)=1,\ \mathrm{ord}_7(x)=-1$ and $\mathrm{ord}_p(x)=0$ for all ...
2
votes
1answer
111 views

$\exp(2)$ does not converge $2$-adically.

We have $\exp(2)= \sum_{i=0}^n {\frac{2^n}{n!}}$. I am trying to show that $\exp(2)$ does not converge $2$-adically. i.e. I need to show $\nu_2 (\frac{2^n}{n!})$ does not tend to $\infty$ as $n\to ...
0
votes
0answers
50 views

Solving matrix equations over $p$-adic rings

Let $F/\mathbb{Q}_p$ be a finite extension, and let $\mathcal{O}_F$ be its ring of integers. Now for $ 0< i,j \le r$ let $B_{i,j} \in \mathit{Mat}_{n \times n}(\mathcal{O}_F)$ be some matrices and ...
6
votes
1answer
108 views

Projective limit involving p-adic numbers

Let $p$ and $q$ be distinct primes. What is the projective limit $$\varprojlim \mathbb R^2 / (p^n \mathbb Z \times q^n \mathbb Z)?$$ That's an exercise from Robert's book A Course in p-adic ...
1
vote
3answers
62 views

$3$-adic expansion of $- \frac{9}{16}$

I get the $3$-adic expansion to be $1+1 \cdot 3+2 \cdot 3^2 +2 \cdot 3^3 + 0 \cdot 3^4+\cdots$. I'm trying to work out a pattern of the coefficients and think it is $1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, ...