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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What is a group of type $(2,2)$?

This is a statement in Serre's A Course in Arithmetic (p. 18). If $p\ne2$, the group $\mathbb{Q}_p^*/\mathbb{Q}_p^{*2}$ is a group of type $(2,2)$. What is a group of type $(2,2)$?
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By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?

this is a question from a book I'm struggling with, please could you provide a clear proof Consider the sequence of rational numbers $a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1 ...
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266 views

For what primes $p$ does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?

this is a question from a book I'm struggling with, please could you provide a clear proof For what primes p does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically? kind thanks
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For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically and, when it does, to what limit?
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p- adic isomorphims

Let $m_p=\{ x \in \mathbb{Q} : v_p(x)>0\}$, $v_p$ $p$-adic valuation. Let $k >0 $ a integer and the ideal $I=m_p^k$. Then, $$O_p/ I \cong \mathbb{Z}/p^k \mathbb{Z} $$ with, $O_p=\{ x \in ...
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Solution of $x^2 = 2$ in $\mathbb{Q}_p$

Let $p \neq 2$. Show that the equation $$x^2=2 \quad (*)$$ has a solution $x \in \mathbb{Q}_p$ iff exists $y \in \mathbb{Z}$ such that $y^2 \equiv 2 \, \pmod p$. $\Rightarrow$ Let $x= y + x_1 p + x_2 ...
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p-adic expansion

Let $x \in \mathbb{Z}_p$ and $\{x_n\}$ such that, $x_n \equiv x_{n+1} (mod \, p^{n+1})$; $0 \leq x_n \leq p^{n+1}-1$ ; $|x-x_n|_p \rightarrow 0, n \rightarrow \infty$. ...
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p-adic norm, Sets equal

Let $p$ a prime, and the p-adic norm $|x|_p = (\frac{1}{p})^{v_p(x)}$, with $v_p$ the p-adic valuation. Show that $$\{ |x|_p : x \in \mathbb{Q}_p \} = \{ p^k : k \in \mathbb{Z}\}$$ My question is ...
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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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P-adic numbers complete/incomplete

P-adic numbers are complete in one sense and incomplete in another sense. Is it so? Firstly, does not complete mean connected? I read somewhere that there is not intermediate value theorem for ...
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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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152 views

Why is the Euclidean metric called the prime at infinity?

I've been studying p-adic analysis recently and after a bit of searching on the web, I haven't found an answer as to why the Euclidean metric is referred to as the 'prime at infinity', and given the ...
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580 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 ...
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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, ...
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Questions regarding p-adic expansion and numbers

As opposed to real number expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base $p$, $p$-adic numbers may expand to the left forever, a property ...
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Spectrum theorem for p-adic matrix analysis

Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way: 1: whether there exist spectrum theorem for p-adic matrix.\ 2: ...
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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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152 views

Prove that the sequence satisfying $a^{p^n}\equiv a^{p^{n-1}}$mod $p^n$ for $n\geq 1$, converges to the Teichmuller Representative

Prove that for any $a\in\mathbb{Z}_p$, the sequence satisfying $a^{p^n}\equiv a^{p^{n-1}}$mod $p^n$ for $n\geq 1$, converges to the Teichmüller Representative congruent to $a$ mod $p$. So a p-adic ...
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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 ...
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Necessary and sufficient conditions for Hensel lifting in the multidimensional case

in Multidimensional Hensel lifting, @Hurkyl gave a neat sufficient condition for the existence of $p$-adic liftings in the multidimensional case. I have finally gotten around (but please also see ...
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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 ...
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Is the set of integers with respect to the p-adic metric compact?

Given the integers and a prime $p$. I thought I had successfully shown that $\mathbb{Z}$ was compact with respect to the metric $|\cdot |_p$, by showing that the open ball centered at zero contained ...
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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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221 views

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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269 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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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 ...
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267 views

Roots of unity in $\mathbb Q_p$

Is there a way to get the number of solutions of an equation like $x^n=1$ in the p-adic field $\mathbb Q_p$, where $p$ is a prime and $n$ a positive integer? I know that for $n=p-1$ there are $p-1$ ...
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If $(x_n)_{n\geq 1}$, $x_n \in \mathbb{Q}$ is p-adic Cauchy, show ord$_p(x_n)$ eventually constant.

Number Theory 1: Fermat's Dream asks the reader to verify the following. They then use this to extend the definition of Ord$_p$ to $\mathbb{Q}_p$. Let $p$ be prime and $a\neq 0$. If ...
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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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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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Set of locations where the Hilbert symbol is not equal to $1$

Let $V$ be the set of prime together with the symbol $\infty$. For a prime $v=p$, denote the $p$-adic numbers by $\mathbb{Q}_p$ and the real numbers by $\mathbb{Q}_\infty$. For $v\in V$ the Hilbert ...
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When a number is a square in the p-adic rationals - proof question (Quadratic Residues)

I'm a little stuck with the proof of a theorem I'm trying to understand. The theorem is as follows: "For odd prime $p$, suppose for $\alpha \in Q_{p}$ (the p-adic rationals) that $|\alpha|_p=1$. Then ...
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Preservation of being a norm under field extension

I'm reading a paper that purports to prove the proposition: Let $K/E$ be a cyclic extension of CM number fields of degree p (an odd prime number). Let $G$ be the Galois group. Let $t$ be the number ...
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Valuation rings of complete non-archimedean fields which are not local

I would like to know how are the valuation rings of complete non-archimedean fields which are not local. Take, for example, $\mathbb{C}_p$, and its valuation ring, ...
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277 views

Roots of 1 in $\mathbb Q_p$

How to prove, that all roots of 1 in $\mathbb Q_p$ are roots of $x^{p-1}-1$? If we consider the ring homomorphism $$ \mathbb Z_p \to \mathbb F_p^*, $$ then we see, that all the roots in power $p-1$ ...
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A $p$-adic “Jacobi sum” with an unramified character

Working over a $p$-adic field with absolute value $|\cdot|$, let $\chi$ be a character on ${\mathfrak o}^\times$ with conductor $n\ge 1$, meaning that $n$ is the smallest integer such that $\chi$ is ...
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Are the p-adic integers the ring of integers of the field of p-adic numbers?

This question was much simpler, but as I was typing it, it became a chain of questions. My starting question was Is $\mathbb{Z}_p$ (obtained by the inverse limit procedure with the directed ...
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4answers
431 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 ...
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$p$-adic unit for $p\neq 2$ is a square in $\mathbb{Z}_p$ if and only if its first digit is quadratic residue modulo $p$.

I heard the statement from the title of this question. $\newcommand{\of}[1]{\left(#1\right)}$ $\newcommand{\df}{\mathrel{\mathop:}=}$ So if I am not totally confused, this formalizes to: Let ...
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1answer
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Quasi-linear time fully homomorphic encryption using p-adic ring homomorphism

I recently encountered a breakthrough in FHE crypto, which claims to have a literally quasi-linear time FHE without any "lambda" factor in the keys and no noise in the cipher-text. This fully ...
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1answer
141 views

Cyclotomy in extensions of $\mathbb{Q}_p$

Let $p$ be a prime number and $K$ be a finite extension of $\mathbb{Q}_p$. Denote by $\zeta_{p^n}$ a primitive $p^n$-th root of unity (where $n$ is a positive integer). Assume that $K$ contains ...
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hensel's lemma for the prime 2

Hensel's lemma for odd primes, as I understand, allows you to determine solutions to various homogeneous quadratic equations modulo $p^n$ by solving an appropriate equation mod $p$ and lifting up. ...
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1answer
71 views

Powers of $\Bbb Z_p\cap \Bbb Q$ on $1+x\Bbb F_q[[x]]$

Set $q=p^r$ for the finite field $\Bbb F_q$. In the formal power series ring $\Bbb F_q[[x]]$, there is a notion of convergence given by the underlying $(x)$-adic topology. If ...
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1answer
171 views

Galois group of maximal $p$-extension of $\mathbb{Q}_l(\zeta_p)$?

Let $F$ be the field obtained by adjoining to $\mathbb{Q}_l$ a $p$-th root of unity, with $p \not = l$. Denote by $F(p)$ the maximal $p$-extension of $F$, i.e. the maximal extension $L:F$ such that ...
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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 ...
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263 views

continued fraction expression for $\sqrt{2}$ in $\mathbb{Q_7}$

Hensel's lemma implies that $\sqrt{2}\in\mathbb{Q_7}$. Find a continued fraction expression for $\sqrt{2}$ in $\mathbb{Q_7}$
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Structure of $\mathbb{Q}_p(\zeta_p)$

Let $p \ne 2$ be prime number and denote by $\zeta_p$ the p-th root of unity. It's well known that $K = \mathbb{Q}_p(\zeta_p)$ has $t=1 - \zeta$ as prime element (generator of the Ideal $P_K = \{ x\in ...
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Inverse mapping theorem over a complete non-archimedean field

Let $K$ be a complete field with respect to a non-trivial non-archimedean absolute value $|\cdot|$. Let $E$ be a vector space over $K$. A norm $||\cdot||$ on $E$ is a map $E \rightarrow \mathbb{R}$ ...
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289 views

About $p$-adic Topology on $\mathbb{Z}$

I want some notes about $p$-adic Topology and some properties with proves about $\mathbb{Z}$ with this Topology. Thank you.
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1answer
113 views

Why $\mathbb{Z}$ with $p$-adic topology is precompact?

Why $\mathbb{Z}$ (group of integer numbers) with $p$-adic topology is a countable precompact metric group with a linear topology? Note : Call a topological group $G$ linear (and its topology a linear ...