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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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 $a \in \mathbb{N}$, ...
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364 views

Isotropy over $p$-adic numbers

Over what $p$-adic fields $\mathbb{Q}_p$ is the form $\langle3, 7, -15\rangle$ isotropic?
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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.
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2answers
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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 ...
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$n$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 ...
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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 ...
4
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264 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 ...
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730 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 ...
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What is the index of the $p$-th power of $\mathbb Q_p$ in $\mathbb Q_p$

In this book it is listed as an exercise to compute the index $[\mathbb Q_p:\mathbb Q_p^p]$. This exercise is appended to a section concerning the structure of unit-group filters, investigating some ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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108 views

Direct proof of compactness of $\mathbb{Z}_p$

Let $\mathbb{Z}_{p}$ be completion of $\mathbb{Z}$ with respect to $p-$norms. Actually I know that $\mathbb{Z}_{p}$ is bijective to Cantor set, which is compact, therefore by homeomorphism, it is also ...
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1answer
166 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 $C_3$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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1answer
61 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 ...
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2answers
320 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 ...
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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 ...
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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3answers
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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 ...
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1answer
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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 ...
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1answer
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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}, ...
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1answer
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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.
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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 ...
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1answer
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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 ...
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2answers
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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\{ ...
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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 ...
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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 ...
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Finding root using Hensel's Lemma

Hensel's Lemma calculates root of a polynomial $\in \mathbb{Z}_p[X]$ but is there any other significance to other branches of mathematics or outside mathematics? Why is finding root of ...
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4answers
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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 ...
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1answer
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Multiplicative Haar measure on $\mathbb{Q}_p$?

I have read in a book that if one takes $\mu$ to be the additive Haar measure on $\mathbb{Q}_p$, the p-adic rationals, then $$\nu(A) := \int_{A} 1/|x|_p dx$$ is a multiplicative Haar measure on ...
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1answer
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Is there a $p$-adic version of Liouville theorem?

That is, if a function $f$ is analytic and bounded in all $K$, a $p$-adic field (or more generally a complete non-archimedean field), has to be constant? And does the theorem work for functions on ...
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3answers
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Visualizing Balls in Ultrametric Spaces

I've been reading about ultrametric spaces, and I find that a lot of the results about balls in ultrametric spaces are very counter-intuitive. For example: if two balls share a common point, then one ...
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Raising a rational integer to a $p$-adic power

Let $p$ be a prime number and $d$ a positive integer. If $n\in \mathbb{Z}_p$ (the ring of $p$-adic integers) how would one define $d^n$? Under what conditions would this be an element of ...
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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" ...
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1answer
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Relationship Between Ring of Integers of a Number Field to P-adic integers

Suppose we have a number field $K$ with ring of integers $\mathcal O_K$. Let $\frak p$ be a prime ideal in $K$ lying over $p\in \mathbb Q$. Then, using the $\frak p$-adic norm, we may define the ...
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1answer
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$t$-adic topology (on $\mathbb F_p(1/t)$)

Recently I found this interesting discussion about algebraically closed fields of positive characteristic. In the answer marked as the top answer, I read about the $t$-adic topology. The $t$-adic ...
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Prove that $\displaystyle\mathbb{Q}_p$ always contains $p$ solutions to the equation $x^p-x=0$

Prove that $\mathbb{Q}_p$ always contains $p$ solutions $a_0,a_1,...,a_{p-1}$ to the equation $x^p-x=0$ satisfying $a_j\equiv j$ (mod p). This is an intense problem as far as I'm concerned. I'm not ...