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

3
votes
1answer
841 views

Characterization of integers which has a $2$-adic square root

Does anyone know an "elementary" proof of the following theorem? Let $k \neq 0$ be a rational integer. Then $k$ admits a square root in $\mathbb{Z}_2$ if $k = 4^a (8b+1)$ for some $a \in \mathbb{N}$, ...
5
votes
2answers
194 views

Irrational P-adics

$\mathbb{Q}_p$ is completion of $\mathbb{Q}$ by defining a new metric. So, with respect to this new metric they are complete. I just want to be sure, are there p-adic rationals? If there are P-adic ...
3
votes
2answers
451 views

Isotropy over $p$-adic numbers

Over what $p$-adic fields $\mathbb{Q}_p$ is the form $\langle3, 7, -15\rangle$ isotropic?
16
votes
3answers
1k views

An automorphism of the field of $p$-adic numbers

Is an automorphism of the field $\mathbb{Q}_p$ of $p$-adic numbers the identity map? If yes, how can we prove it? Note:We don't assume an automorphism of $\mathbb{Q}_p$ is continuous.
15
votes
4answers
590 views

Divergent series and $p$-adics

If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct. Surely ...
7
votes
2answers
988 views

$n$th powers in the p-adics

Suppose $K$ is a $p$-adic field (finite extension of the $p$-adics), and let $n$ be any integer (independent of what $p$ is). Define $U$ to be the set of all $x$ in $K$ such that $|x| = 1$ and such ...
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 ...
10
votes
2answers
844 views

The p-adic numbers as an ordered group

So I understand that there is no order on the field of p-adic numbers $\mathbb{Q_p}$ that makes it into an ordered field (i.e.) compatible with both addition and multiplication. Now, from the ...
4
votes
1answer
397 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, ...
33
votes
3answers
3k views

How far are the $p$-adic numbers from being algebraically closed?

A few days ago I was recalling some facts about the $p$-adic numbers, for example the fact that the $p$-adic metric is an ultrametric implies very strongly that there is no order on $\mathbb{Q}_p$, as ...
11
votes
6answers
362 views

Can one show a beginning student how to use the $p$-adics to solve a problem?

I recently had a discussion about how to teach $p$-adic numbers to high school students. One person mentioned that they found it difficult to get used to $p$-adics because no one told them why the $p$-...
16
votes
1answer
417 views

what are the p-adic division algebras?

Is there a classification of division algebras over $\mathbb{Q}_p$? There are field extensions of $\mathbb{Q}_p$, but are there any others? In particular, I want to know if they are all commutative. ...
4
votes
1answer
109 views

sequence $\{a^{p^{n}}\}$ converges in the p-adic numbers.

Let $a\in \mathbb{Z}$ be relatively prime to $p$ prime. Then show that the seqeunce $\{a^{p^{n}}\}$ converges in the $p$-adic numbers. This to me seems very counter intuitive. Since $(a,p)=1$ the ...
4
votes
1answer
304 views

Multidimensional Hensel lifting

I have a question about a practical application of (some) generalised form of Hensel's Lemma. I cannot find it stated in an appropriate form in Bourbaki or anywhere else, so here goes ... Let $p$ be ...
7
votes
2answers
211 views

Does $\log p$ make sense in a finite extension of $\mathbb{Q}_q$?

Can we make sense of the logarithm of prime in some algebraic extension of $\mathbb{Q}_q$, where either $q \neq p$ or $p = q$ and both prime numbers? Some reflections: A naive starting point is ...
6
votes
2answers
1k views

Why are closed balls in the $p$-adic topology compact?

I was skimming through some of this paper Measurable Dynamics and Simple $p$-adic Polynomials out of curiosity. A few pages in, the author claims that closed balls are both open and compact sets in ...
4
votes
3answers
631 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
178 views

Valuations on number fields

I'm trying to explicitly compute modular representations of some finite groups -- the easiest example to discuss is the cyclic group $C_3$ when $p=3$. The three ordinary irreducible modules for $C_3$, ...
3
votes
1answer
116 views

Why doesn't the equation have a solution in $\mathbb{Q}_2$?

I have to find for which primes $p$, the equation $x^2+y^2=3z^2$ has a rational point in $\mathbb{Q}_p$. According to my notes: Obviously, $\forall p \in \mathbb{P}, p \nmid 2 \cdot 3$, there is a ...
11
votes
1answer
320 views

Local solutions of a Diophantine equation

I am trying to prove that the equation $$3x^3 + 4y^3 +5z^3 \equiv 0 \pmod{p}$$ has a non-trivial solution for all primes $p$. I am sure that this is a standard exercise, and I have done the easy parts:...
1
vote
2answers
145 views

$5$-adic expansion of $−2$

Let $\alpha = \sum_{n=0}^{\infty} a_n\ p^n$ be the expansion of a p-adic unit. So $0<a_0<p$ and $0\leq a_n<p$ for $n \geq 1$. Show that $\beta = -\alpha$ has the expansion $\beta = \sum_{...
1
vote
2answers
67 views

Finite index subgroup $G$ of $\mathbb{Z}_p$ is open.

Suppose $[\mathbb{Z}_p:G] = n <\infty$. Write $n = p^km$ with $p\nmid m$. The idea is to show that $p^k\mathbb{Z}_p = n\mathbb{Z}_p \subseteq G$, after which I am done, since for any $x\in G$ we ...
-3
votes
1answer
237 views

Is 0.9999… equal to -1?

I read the following in a wikipedia article about 0.9999...: A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that $0.999... = 1$ but was ...
18
votes
0answers
249 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^{\...
10
votes
2answers
993 views

Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...
18
votes
4answers
351 views

Dedekind-Like construction of p-adic numbers

Recently I've been studying p-adic numbers. I understand the idea of a cauchy completion of the rationals with respect to the metric defined by the norm $\vert\vert \cdot \vert \vert_p $. When I was ...
9
votes
1answer
814 views

Is $\mathbb Q_r$ algebraically isomorphic to $\mathbb Q_s$ while r and s denote different primes?

It is obvious that $\mathbb{Q}_r$ is topologically isomorphic to $\mathbb Q_s$ while $r$ and $s$ denote different primes. But I really don't know whether it is true in the aspect of algebra. As I ...
9
votes
2answers
425 views

Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$

The following is a historical question, but first some background: Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
7
votes
5answers
539 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 ...
7
votes
1answer
200 views

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1 and $p$ is an odd prime.
4
votes
1answer
92 views

Which $p$-adic fields contain these numbers?

Question: Determine the $p$-adic fields which contain $$ a)\;\sqrt{-1} \qquad b)\;\sqrt{3} \qquad c)\;\sqrt{-7} \qquad d)\;\sqrt{17}$$ I have no idea on this as I am completely confused with $p$-...
12
votes
2answers
364 views

Why is $O_K\otimes \mathbb{Z}_p\cong \oplus_{\mathfrak{p}|p}O_{K,\mathfrak{p}}$?

In my old number theory notebook this is stated as a fact. However, I ran into problems when I tried to prove it. First let me state the (supposed) theorem accurately: Theorem (?) Let $K$ be a ...
8
votes
2answers
1k views

Is the algebraic closure of a $p$-adic field complete

Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?) Why is (or why isn't) an algebraic closure $\overline{K}$ complete? Maybe this holds more ...
6
votes
0answers
106 views

A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
7
votes
2answers
116 views

Existence of $\sqrt{-1}$ in $5$-adics, show resulting sum is convergent.

I know that to prove the existence of a square root of $-1$ in $\mathbb{Z}_5$, I can just plug $x = -5$ and $a = 1/2$ into the Taylor expansion$$(1 + x)^a = \sum_{n=0}^\infty \binom{a}{n} x^n.$$...
5
votes
1answer
1k 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 ...
4
votes
1answer
84 views

Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of $\...
4
votes
1answer
314 views

Why exponential function on p-adic numbers is meaningless?

In the notes, page 3, it is said that $e^{2\pi i r y}$ is meaningless if $y$ is a general p-adic number. Why exponential function on p-adic numbers is meaningless? Thank you very much.
4
votes
1answer
145 views

Unramified extension is normal if it has normal residue class extension

Let $K/F$ be an unramified extension such that $\rho_K / \rho_F$ (the corresponding extension of residue classes) is normal. Prove $K/F$ is normal. I guess I need to do some polynomial lifting, but ...
2
votes
1answer
174 views

Is $123456788910111121314\cdots$ a $p$-adic integer?

On the back of this question comes the natural question of whether the string $$1234567891011121314\!\cdots$$ is even a number at all. While that sort of question is vague, given the lack of generic ...
7
votes
1answer
140 views

A valuation-like function $w: \mathbb{N}^{+} \rightarrow \mathbb{N}$ is a $p$-adic valuation?

This question is a variant of problem 4, pg. 21, from Birkhoff and Maclane, A Survey of Modern Algebra. Given a function $w: \mathbb{N}^+ \rightarrow \mathbb{N}$ that behaves like a valuation ...
5
votes
3answers
283 views

Finding $p^\textrm{th}$ roots in $\mathbb{Q}_p$?

So assume we are given some $a\in\mathbb{Z}_p^\times$ and we want to figure out if $X^p-a$ has a root in $\mathbb{Q}_p$. We know that such a root must be unique, because given two such roots $\alpha,\...
5
votes
1answer
294 views

At which p-adic fields does the equation have no solution?

I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't have, I have to find at which p-adic fields it has ...
5
votes
1answer
225 views

Properties of squares in $\mathbb Q_p$

Let $\mathbb Q_p$ be the field of $p$-adic numbers. I know that for $p \neq 2$ an element $x=p^n u \in \mathbb Q_p^\times$ (with $n \in \mathbb Z$ and $u \in \mathbb Z_p^\times$) is a square if and ...
5
votes
2answers
616 views

Unramified p-adic extension implies Galois

I am looking for a short proof that if $L \supset K$ are finite extensions of the p-adic numbers $\mathbb{Q}_p$, then if $L/K$ is unramified, $L/K$ is Galois. I think the proof is related to somehow ...
5
votes
2answers
114 views

Show that $y^2=x^3+1$ has infinitely many solutions over $\mathbb Z_p$.

I first compared it with how I would solve this over the real numbers. You would say: $y^2=\alpha$ has a solution for all $\alpha>0$, of which there are infinitely many. $x^3+1>0$ for all $x&...
4
votes
1answer
309 views

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}, ...
3
votes
2answers
258 views

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 $...
3
votes
4answers
659 views

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 ...
2
votes
2answers
80 views

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\{ ...