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.

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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 ...
18
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1answer
258 views

Why is there “no analogue of $2i\pi$ in $\mathbf C_p$”?

In his paper Fonctions L p-adiques, Pierre Colmez says: Tate a montré qu'il n'existait pas dans $\mathbf C_p$ d'analogue $p$-adique de $2i \pi$ et donc par conséquent que les périodes $p$-adiques ...
14
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3answers
466 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.
13
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1answer
301 views

Are all $p$-adic number systems the same?

After just having learned about $p$-adic numbers I've now got another question which I can't figure out from the Wikipedia page. As far as I understand, the $p$-adic numbers are basically completing ...
13
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1answer
197 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. ...
13
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4answers
200 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 ...
12
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2answers
350 views

Tensor products of p-adic integers

These are relatively simple questions, but I can't seem to find anything on tensor products and p-adic integers/numbers anywhere, so I thought I'd ask. My first question is: given some ...
11
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1answer
322 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 ...
11
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2answers
300 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 ...
10
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1answer
105 views

Show that $x^4+1$ is reducible in p-adic numbers $\mathbb{Q}_p$ for p>2 prime.

This is a homework problem for algebraic number theory but I'm having trouble getting started. Do I use induction in general, or show this holds for $p \equiv 1,3$ (mod 4)? Any help would be ...
10
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1answer
285 views

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in ...
10
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1answer
256 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 ...
9
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6answers
191 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 ...
9
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4answers
429 views

Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?

I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this: Show that $x^2-82y^2=\pm2$ has solutions in every ...
9
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2answers
658 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 ...
9
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2answers
375 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 ...
9
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347 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 ...
9
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1answer
125 views

diagonalizing a matrix over the $\ell$-adics

Let $M$ be a $2 \times 2$ matrix with coefficients in $\mathbb{Z}_{\ell}$ whose characteristical polynomial is $$ P(T) = T^2- (a+d) T + (ad-bc). $$ I've encountered the following assertion: If ...
9
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0answers
123 views

p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie ...
8
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2answers
241 views

p-adic numbers and $\mathbb{F}_p$

As you probably know, there is a morphism of rings : $\mathbb{Z}_p\longrightarrow \mathbb{Z}/p\mathbb{Z}$ which sends a formal sum $\sum_{i\geq 0}a_ip^i$ to $a_0$ (here $\mathbb{Z}_p$ is the ring of ...
8
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2answers
562 views

Are all nonarchimedean valuations discrete?

I am studying valuation theory on the way to local class field theory, and the texts I have looked at immediately focus on discrete valuations in developing the theory of nonarchimedean valuations. ...
8
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1answer
318 views

Why $p$-adically interpolate?

I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
8
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656 views

Binomial coefficients: how to prove an inequality on the $p$-adic valuation?

In section 4 of the article by Afred van der Poorten's A Proof That Euler Missed ... the following inequality is used: $$\nu_{p}\displaystyle\binom{n}{m}\leq\left\lfloor\dfrac{\ln n}{\ln ...
8
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1answer
451 views

Structure of p-adic units

I am trying to understand the structure of the $p$-adic units. I know that we can write $$\mathbb{Z}_p^\times \cong \mu_{p-1} \times 1 + p\mathbb{Z}_p,$$ where $\mu_n$ are the $n$th roots of unity in ...
8
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1answer
115 views

Does there exist an ordered ring, with $\mathbb{Z}$ as an ordered subring, such that some ring of p-adic integers can be formed as a quotient ring?

I'm not sure if this might instead be MathOverflow material, but I'll give it a shot here first anyway. Does any ring $R$ exist that satisfies the following properties? $R$ is a totally ordered, ...
8
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1answer
141 views

$\mathbb Z_p$-rank $\leq r_1+r_2-1$ in Leopoldt's conjecture

I am trying to understand Leopoldt's conjecture as formulated in section 5.5 of Washington's "Introduction to Cyclotomic Fields". For an algebraic number field $K$ let $E$ denote the global units, ...
8
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1answer
221 views

$S$-Units notation and Dirichlet's unit theorem

I'm having a hard time understanding some notions of a paper I'm working on. Let $L/K$ be a finite normal extension of number fields and $S$ be a set of places of $K$ prime to $p$ where $p$ denotes an ...
7
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4answers
129 views

How do you prove that $\Bbb{Z}_p$ is an integral domain?

Let $\Bbb{Z}_p$ be the $p$-adic integers given by formal series $\sum_{i\geq 0} a_i p^i$. I'm having trouble proving that it's an integral domain.
7
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2answers
592 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 ...
7
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5answers
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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
2answers
1k views

What do the $p$-adic roots of unity look like?

I know that $\mathbb{Z}_p$ has all the $p-1^{st}$ roots of unity (and only those). Is it true that mod $p$ they are all different? Meaning, is the natural map $\mathbb{Z}_p \rightarrow \mathbb{F}_p$, ...
7
votes
2answers
567 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 ...
7
votes
2answers
323 views

Roots of unity in $\mathbb{Q} _{11}$

Here $\mathbb{Q} _{11}$ denotes the 11-adic field. How can I show that the only root of unity of order 7 in this field is 1? Is it true that for any two distinct primes $p,q$, the only root of unity ...
7
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1answer
98 views

Local solutions over $\mathbb{Q}_p$ but no solutions over $\mathbb{Q}$

I was looking at a set of notes that states the equation $x^4-17=2y^2$ is solvable locally over $\mathbb{Q_p}$ for every $p$ , but is not solvable over $\mathbb{Q}$. Now, this is not a homework ...
7
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3answers
299 views

A property of some sequences of natural numbers (and their binary representation)

Let's consider a sequence of natural numbers $a_n$, represented in binary, with the following properties: $\forall n \in \mathbb{N}$ the number $a_n$ is represented with $n$ binary digits $\forall ...
7
votes
2answers
312 views

What does the Haar measure on $\hat{\mathbf{Z}}$ look like?

What does the Haar measure on $\hat{\mathbf{Z}} = \prod_p \mathbf{Z}_p$ look like? Does it bear any relation to the "upper density" of a subset $S\subset\mathbf{N}$ defined by $m(S) = ...
7
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1answer
184 views

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}$.

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}=\{\sum_{i=0}^\infty a_ip^i:0\le a_i \le p-1\}$. I was trying ...
7
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1answer
111 views

Problems that are made easy by using p-adic numbers

Does anybody know of elementary problems that can be be solved using the p-adics? Solutions are preferred.
7
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1answer
121 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 ...
7
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0answers
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Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
6
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4answers
331 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 ...
6
votes
3answers
193 views

Tensor product of a number field $K$ and the $p$-adic integers

In a paper by J. Jones and D. Roberts (http://math.la.asu.edu/~jj/localfields/database.pdf) we are introduced to an isomorphism $K \otimes \mathbb{Q}_p \cong \prod\limits_{i=1}^g K_{p,i}$, where $K$ ...
6
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2answers
152 views

Does $\mathbb{Q}_p \cap \overline{\mathbb{Q}}$ depend on the embedding of $\overline{\mathbb{Q}}$ into $\overline{\mathbb{Q}_p}$?

I think the title pretty much says it all. I'm getting confused in the subtle parts of a proof, and would appreciate some help.
6
votes
2answers
181 views

Easy way to determine the primes for which $3$ is a cube in $\mathbb{Q}_p$?

This is a qual problem from Princeton's website and I'm wondering if there's an easy way to solve it: For which $p$ is $3$ a cube root in $\mathbb{Q}_p$? The case $p=3$ for which $X^3-3$ is not ...
6
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1answer
294 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
6
votes
3answers
176 views

When is a $p$-adic unit an $m$-th power

Given $x\in\mathbb{Q}_p^*$, we can write $x=p^nu$ where $n\in\mathbb{Z}$ and $u\in\mathbb{Z}_p^*$. Then we can decide whether or not $x$ is a square by looking at $n$ and $u$. If $p\neq 2$ then $x$ is ...
6
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1answer
53 views

Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
6
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1answer
185 views

Line in a proof on p69 in Cassel's Local Fields

I'm trying to read the proof of LEMMA 6.1 (Nagell) Let $u_n$ be defined by $u_0=0$, $u_1=1$ and $u_n=u_{n-1}-2u_{n-2} \hspace{20pt} (n\geq 2)$. Then $u_n=\pm1$ only for $n=1,2,3, ...
6
votes
1answer
105 views

What does root of unity in $\mathbb{Z}_p$ look like?

Let $p$ be an odd prime. Then by Hensel's lemma it's clear that $\mathbb{Z}_p $ contains all $p-1$th root of unity which reduces to $1$, $2$, ... , $p-1$ in $\mathbb{F}_p$. My question is do we know ...
6
votes
1answer
111 views

Relations between irreducibility on $\mathbb{Q}[x]$, and on $\mathbb{Q}_p[x]$ ($p$-adic numbers)

I'm reading "$p-$adic numbers: An introduction" by Fernando Q.Gouvêa, and I'm currently on page 79 of the book. Problem 121. Show that the equation $(X^2 - 2)(X^2 - 17)(X^2 - 34) = 0$ has a root ...