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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On the numerators of Bernoulli numbers

Von Staudt-Clausen theorem implies that $pB_{2n} \in \mathbb{Z}_{p}$ for all primes $p$ and for all $n \in \mathbb{N}$. It means that the highest power of any prime that can occur in the denominator ...
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379 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 $a \in \mathbb{N}$, ...
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176 views

Are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic?

If $p$ and $q$ are distinct prime number, are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic as topological space?
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141 views

A non-Archimedean norm definition can be strengthened.

See Andrew Baker's p-adic notes: For a non-Archimedean norm $N$ it is true that "$N(x + y) \leq \max\{N(x), N(y)\}$, with equality if $N(x) \neq N(y)$." Having trouble proving this. Please ...
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How can every $p$-adic integer be the limit of a sequence of non-negative integers?

See Andrew Baker's P-adic Notes. Every element of $\mathbb{Z}_p = \{a \in \mathbb{Q}_p : |a|_p \leq 1 \}$ is a limit of a sequence of non-negative integers, with respect to the $|\cdot|_p$ norm. How ...
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p-adic expansion of a rational number

Studying $p$-adic numbers I encountered the following theorem: Given a eventually periodic sequence $(a_n)_{n=k}^{\infty}$ such that $0 \le a_n <p$, the sum \begin{equation*} ...
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178 views

Inverse limit of $\mathbb{Z}/(n!)\mathbb{Z}$

I am interested to know if there is a standard name for the inverse limit, $\hat{\mathbb{Z}}_!$, say, of the inverse system of rings $$\ldots \rightarrow \mathbb{Z}/((n+1)!)\mathbb{Z} \rightarrow ...
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Finitely generated submodule of $p$-adic module is direct summand? [closed]

Are finitely generated submodules of a $p$-adic module direct summands?
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63 views

arithmetic of p-adics

I need only a fast check if the following expressions are correct. Let $\frac{a}{b} \in \mathbb{Q}\def\ord{\operatorname{ord}}$ then the following fact olds $|x|_p = \left|\dfrac{a}{b}\right|_p = ...
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118 views

Is $(\mathbb Z/p^{n+1}\mathbb Z)^\times\cong (\mathbb Z/p\mathbb Z)^\times\times(\mathbb Z/p^n\mathbb Z)$?

Let $p$ be an odd prime number and $n$ any positive integer. Is $(\mathbb Z/p^{n+1}\mathbb Z)^\times\cong (\mathbb Z/p\mathbb Z)^\times\times(\mathbb Z/p^n\mathbb Z)$ as groups? This seems very ...
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68 views

Is $p^n\mathbb Z_p\cong \mathbb Z_p$ as additive groups?

Is it true that $p^n\mathbb{Z}_p\cong \mathbb Z_p$ as additive groups? Here $\mathbb Z_p$ is the ring of $p$-adic integers for $p$ prime and $n$ is any positive integer. Thanks
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86 views

If $\,p\,$ is prime, is $\,p^n\mathbb{Z}_p=\mathbb{Z}_p\,$ for any positive integer $n$?

Is $\,p^n\mathbb{Z}_p=\mathbb{Z}_p\,$ for any positive integer $n\,?$ $\mathbb{Z}_p =$ ring of $p$-adic integers, $\,p$ prime. Thanks.
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206 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}, ...
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182 views

A characterization of the module function on a locally compact division ring

References: Weil's Basic Number Theory(written as BNT). Bourbaki's Commutative Algebra(written as BCA). Let $K$ be a topological ring with an identity. Suppose every non-zero element of $K$ is ...
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578 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.
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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 ...
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116 views

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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79 views

p-adic Eisenstein series

I'm trying to understand the basic properties of the p-adic Eisenstein series. Let $p$ be a prime number. Define the group $X = \begin{cases} \mathbb{Z}_p\times \mathbb{Z}/(p-1)\mathbb{Z} & ...
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Proving if $|a|_p=1$ then $a$ is invertible in $\mathbb{Z}_p$

I decided to take a look $p$-adic integers. I am trying to show that $$a \in \mathbb{Z}_p \text{ is invertible if and only if } |a|_p=1$$ where $$|x|_p= \left\{ \begin{array}{ll} p^{-n} & ...
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174 views

$p$ prime, $P = \left\{ \frac{m}{p^e} \middle| m, e\in \mathbb{Z} \right\}$. Prove that $\mbox{Ext}(P; \mathbb{Z}) \cong \mathbb{Z}^{(p)}/\mathbb{Z}$

I don't know why the book Homology by Saunders Mac Lane is wwaaayyy tttoooo hard to digest. :((( This is like the third time I read this book, but still not clear is everything, and to tell the ...
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presentation of the inertia group of $p$-adic fields

It is known (see here) that the absolute Galois group of a $p$-adic number field $K$ (ie a finite extension of $\mathbb Q_p$) is topologically finitely presented for any odd prime $p$. Is it also ...
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On changing from '<' to '$\le$' when taking limits (with norm $|\bullet|_p$)

I'm reading Gouvêa's book on $p-$adic, and there's one problem that I don't think I really get it. Here's a proposition, and the problem attached to it. It's on page 57, 58 of the book. ...
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Lower Bound on Number of Wildly Ramified Extensions of $\mathbb{Q}_p$

Suppose $p$ is prime and $p$ divides $e$ divides $n$. Typically up to isomorphism there are a lot of wildly ramified extensions of $\mathbb{Q}_p$ which have degree $n$ and ramification index $e$. ...
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50 views

Convergence in $\mathbb{Z}_p$

Here is my question: Let $\alpha_0, \dots, \alpha_{p-1} \in \mathbb{Z}_p$ be such that $\alpha_i \equiv i \pmod{p}$ for all $i = 0,\dots, p-1$. Show that, for any $x\in \mathbb{Z}_{p}$, you can find ...
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250 views

An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let K be a field, and let $|\cdot|_{1},\cdots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
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106 views

Is the endomorphism of $\mathbb{Z}_{p}$ induced by multiplication by $p^{n}$ surjective?

Let $p$ be a prime number. Is it true that $p^{n}\mathbb{Z}_{p}\cong\mathbb{Z}_{p}$ as additive groups for any natural number $n$ and if so, why? Here, $\mathbb{Z}_{p}$ denotes the ring of $p$-adic ...
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computation of an $H^2$

Let $p$ be a prime number and $\mathbb{Q}_p$ the field of $p$-adic numbers. Let $G_p$ be the absolute Galois group of $\mathbb{Q}_p$ and fix an absolutely irreducible representation $$ \rho : G_p \to ...
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Discriminant of Finite-Dimensional Extension of $\mathbb{Q}_p$

For an $n$-dimensional extension $K$ of $\mathbb{Q}_p$, we have $K$'s "ring of integers" $\mathcal O_K$ and its uniformizer $\varpi$. We also have the ring of $p$-adic integers $\mathbb{Z}_p$, with ...
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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 ...
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Ring of the residual classes $(\Bbb Z/p\Bbb Z)^\times$? $p$-adic integer?

In a recent question we raised the theorem: for a given prime $p$ and a given power $m$ the representation of any positive integer $n\in \Bbb N$ in the form: $$ n=(a_u p - b_u) \; p^m$$ is unique ...
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81 views

connect p-adic expansion and fundamental theorem of arithmetic?

On the way to explain a $p$-adic expansion, we consider, when dealing with natural numbers, if we take $p$ to be a fixed prime number, then any positive integer expansion in the form can be written as ...
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How to understand the infraconnected set

I begin to study some p-adic analysis. I find it is hard to understand the infraconnected set. Who can give me some examples to show it? Is it relate with the connected set in the topology? I also ...
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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 ...
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142 views

Where do $p$-adic numbers and $p$-Sylow theory both appear?

Both $p$-adic numbers and $p$-Sylow theory are by design "arithmetic" ways of "localizing," so it stands to reason they might be in cahoots in certain contexts. Are they?
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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 ...
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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 $K$ ...
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Image of the Norm on a Finite Dimensional Extension of $\mathbb{Q}_p$

I've been trying to see whether following assertion is true in order to give a quick proof of another problem I was doing: if $K$ is a finite dimensional extension of the $p$-adic numbers ...
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205 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 ...
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Extension of valuation to the algebraic extension of a number field.

I am trying to get the idea how we can extend the $p$-adic valuation on $\mathbb Q$ to an algebraic extension. In particular, how to extend the $p$-adic valuation for $p = 5$ from $\mathbb Q$ to ...
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Why are the p-adic integers a linearly ordered group? [duplicate]

In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group. I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
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Maximal compact subgroup of $GL_n(\mathbb C_p)$

It is known that the general linear group $GL_n(\mathbb Q_p)$ over the $p$-adic numbers has $GL_n(\mathbb Z_p)$ as a maximal compact subgroup and every other maximal compact subgroup of $GL_n(\mathbb ...
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Gelfand's formula, different field

Gelfand's formula says that for a complex matrix $A \in \mathbb{C}^{n \times n}$, $$\rho(A) = \lim_{m \rightarrow \infty} \|A^m\|^{1/m},$$ where $\rho$ is the spectral radius (norm of maximal ...
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Group structure of $\mathbb{Q}_p ^* / \mathbb{Q}_p ^{*3}$

Let p be 1 mod 3 (separate question: work out 2 mod 3). What is the group structure of the abelian group $\mathbb{Q}_p ^* / \mathbb{Q}_p ^{*3}$? $\mathbb{Q}_p ^*$ refers to the group of units in ...
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Prove minimal polynomial over $\mathbb{Q}$ is reducible over $\mathbb{Q}_p$ and $\mathbb{R}$

Let $f$ be the minimal polynomial of $\alpha = \sqrt{-1} + \sqrt{17} + \sqrt{-17}$ over $\mathbb{Q}$ (with degree 4). Prove $f$ is reducible over $\mathbb{Q}_p$ (p-adic rationals) for all primes p and ...
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all various cubic extensions of Q7

I need to classify all various cubic extensions of $\mathbb Q_7$? How can one do it?
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257 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 ...
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198 views

p-adic isomorphism $\mathbb{Q}_p\not\cong \mathbb{Q}_q\iff p\ne q$ [duplicate]

In class I learn that $\mathbb{Q}_3\not\cong\mathbb{Q}_5$ because one of them has $\sqrt{2}$ the other doesn't. Also professor asks us to find reference that $\mathbb{Q}_p\not\cong \mathbb{Q}_q\iff ...
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110 views

How to show that $\mathbb{Q}_p$ cannot be ordered?

I've seen many references on this site and others to the fact that the $p$-adic numbers cannot be ordered, but the closest I've seen to a proof of this is Wikipedia's vague reference to ...
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89 views

How to find $\sup(\{|x-y|_p : x,y\in B(0;r)\})$

Just to clarify the notation and the question: Working in p-adic space $\mathbb{Q}_p$, we have the norm $|x|_p=p^{-ord_p(x)}$ and we define the metric over this space as $d(x,y)=|x-y|_p$. We are ...
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62 views

Irreducible Polynomials over $\mathbb{Q}_p$

I was reading a paper and the author had an irreducible quadratic polynomial $f(x)$, with (non-real) root $\alpha$. He stated that if $p$ ramified or stayed prime in $\mathbb{Q}(\alpha)$ then $f(x)$ ...