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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Are there p-adic manifolds?

Is there anything resembling a manifold on the field of p-adic or complex p-adic fields? If so is there a connection to algebraic geometry as rich as in the reals?
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113 views

non integer p adic expansion (special case)

I need to calculate the 5 adic expansion of $\frac{1}{45}$. Since i cannot compute it normally, i expand $\frac{1}{45}$ into $\frac{1}{5}*\frac{1}{9}$. I calculated the 5 adic expansion of ...
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383 views

P-adic expansion construction

Can anyone teach me about p-adic expansion? especially the case where we have to expand a square root. I need to know how to construct them. for example: the 7-adic expansion of $\sqrt{305}$. This ...
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1answer
494 views

P-adic integers and roots of unity

Show that $\Bbb Z_p$ contains all the $(p-1)$th roots of unity. For which primes $p$ does $\Bbb Z_p$ contains primitive fourth roots of unity? Here $\Bbb Z_p$ is the ring of $p$-adic integers. Proving ...
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90 views

$p$-adic representation and $p$-adic analytic group ($p$-adic Lie group)

A $p$-adic representation of a group $G$ is continuous group homomorphism $$\rho: G \to GL_n(\mathbb{Q}_p)$$ How are those representations related to $p$-adic analytic groups (= $p$-adic Lie ...
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Ring of integers of unramified extension

Let $L/K$ be unramified extension of local fields, and $k,l$ - their residue fields, $l=k(\overline \alpha)$. Is it true that $\mathcal O_L=\mathcal O_K[\alpha]$? And can it be proved if it's true.
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$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$?

Let $K$ be a totally ramified extension of $\mathbb Q_p$ of degree $n$. Then $$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n) .$$ What is this isomorphism?
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84 views

Hensel's lemma in $\mathbb Z_2$

Can you give me a concrete example for a quadratic form $$ f(x,y)=ax^2+bxy+cy^2 \in \mathbb Z_2[x,y] $$ which has a primitive solution $(x^*,y^*) \in \mathbb Z_2 \times \mathbb Z_2$ (mod 4) with the ...
4
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1answer
170 views

Can we determine $A= 1!+2!+3!+…$'s digits starting from last?

After reading a bit about p-adic numbers, I came up with an idea. We know that for every natural number $k$, there exists a natural number $n$ so that for every $m>n$, there are at least $k$ zero ...
4
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1answer
57 views

$\mathbb{Q}^*$ closed in the finite ideles?

I want to consider the (topological) group of 'finite' ideles: If $$\mathbb{A}_\text{fin} = \widehat{\prod}^{\mathbf{Z}_p}_p \mathbf{Q}_p$$ (the 'hat' indicates the so-called restricted product ...
3
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1answer
125 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 ...
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0answers
30 views

Automorphism on $\mathbb{Q}_p$ [duplicate]

How many automorphism are there in the field of $p$-adic? % I suspect that there's only one automorphism, the Identity. But I stuck with the continuously of the automorphism in $\mathbb{Q}_p$.
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1answer
45 views

How to calculate the derivative of a function in $\mathbb{Q}_p$?

On Wikipedia it is stated that the function $$ f:\mathbb{Q}_p\to \mathbb{Q}_p $$ with $f(x)=(1/|x|_p)^2$ if $x\neq 0$ and $f(0)=0$ is differentiable and its derivative is the zero-function. How ...
2
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1answer
75 views

Discrete valuation on $p$-adic numbers

For the ring of $p$-adic integers $\mathbb Z_p$ let $\mu_n: \mathbb Z_p \to \mathbb Z / p^n \mathbb Z$ be the projection mapping. Consider $\mathbb Z / p^n \mathbb Z$ with the discrete topology. Is ...
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2answers
176 views

Binary representation of 2-adic integers

I would like some examples of the binary representation of 2-adic integers that are not standard integers. What is the 2-adic expansion of $1/3$? Of $-1/3$? What number does $...010101$ represent?
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1answer
71 views

Structure of $(\mathbb{Z}/p^k\mathbb{Z})^\times$ other than via $p$-adics?

I think I heard somewhere long ago that $\mathbb{Z}/p^k\mathbb{Z}$'s unit group is cyclic if $p$ is odd and $C_2\times C_{2^{k-2}}$ if $p=2$. I remember trying to prove it and finding it surprisingly ...
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250 views

Ring of $p$-adic integers $\mathbb Z_p$

There are a few ways to define the $p$-adic numbers. If one defines the ring of $p$-adic integers $\mathbb Z_p$ as the inverse limit of the sequence $(A_n, \phi_n)$ with $A_n:=\mathbb Z/p^n \mathbb ...
7
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1answer
124 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 ...
0
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1answer
106 views

Modular 2-adic Integers Question

I would like to know if the following statement is true in the 2-adic integers. $\forall n( n=0 \lor Ex( (x \neq 0 \land x+x=0 \bmod n) \lor (x+x=1 \bmod n) ))$ I will define a modulo predicate as: ...
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Definiton of p-adic integers

Definiton A $p$-adic integer is a (formal) series $$\alpha=a_0+a_1p+a_2p^2+\ldots$$ with $0\leq a_i<p$. The set of $p$-adic integers is denoted by $\mathbb{Z}_p$. If we cut an element ...
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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 ...
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71 views

Basic question about $p$-adic expansions

I was recently introduced to the $p$-adic numbers, and have been asked to show that for $n > 0$, the $p$-adic expansion of $\frac{1}{1 - p^n}$ is $\sum_{i=0}^\infty p^{in}$. Could someone tell me ...
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1answer
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Generators of Zp* and p-adic cyclotomic character

Let $p$ be an odd prime number. It is known that $\mathbb{Z}_p^{\times}$ is topologically cyclic. Now let $\chi_{\mathrm{cyclo}} : Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p) \to \mathbb{Z}_p^{\times}$ be ...
3
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1answer
99 views

Why $\operatorname{Gal}(\mathbb{Q}_q(\zeta_p)/\mathbb{Q}_q)$ is isomorphic to $\mathbb{Z}/f\mathbb{Z}$

I would like to ask a question on this answer. I don't understand why $\operatorname{Gal}(\mathbb{Q}_q(\zeta_p)/\mathbb{Q}_q)$ is isomorphic to $\mathbb{Z}/f\mathbb{Z}$, where $f$ is the degree of the ...
11
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1answer
415 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 ...
3
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0answers
38 views

Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
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1answer
108 views

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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1answer
355 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 $a \in \mathbb{N}$, ...
3
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1answer
171 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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1answer
133 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 ...
3
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1answer
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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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3answers
601 views

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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2answers
170 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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60 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 = ...
2
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1answer
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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2answers
65 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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2answers
85 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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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
178 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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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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3answers
113 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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1answer
75 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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2answers
173 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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0answers
45 views

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 ...
2
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2answers
53 views

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. ...
7
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1answer
123 views

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 $e$. ...
2
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1answer
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 ...