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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An equation is solvable in $\mathbb Z_p$

Suppose $p$ is an odd prime and $a\in \mathbb Z_p$ with $a\equiv 1 \mod p^2 $. Then $x^p=a$ is solvable over $\mathbb Z_p$. I want to prove $x^p\equiv a\mod p^v$ is solvable for arbitrary $v\geqslant ...
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56 views

Polynomials generating the same $p$-adic fields

I wonder if the following fact is true: Pick $l\in \mathbb N$ a number and let $f,g\in \mathbb Z_p[x]$ be monic polynomials with coefficients in the rings of $p$-adic integers such that $f\equiv g \...
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$\mathbb{Z}_p$-extensions of CM-fields

I am trying to prove some consequences of Iwasawa's Theorem for CM-fields. There is a sequence of CM-fields $$K=K_0\subseteq K_1 \subseteq \dots \subseteq K_\infty$$ so that $K_\infty/K$ is a $\mathbb{...
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$\mathbb{Z}_p$ is an Integral Domain

Assume that we define the ring of the $p$-adic integers as the projective limit $$\mathbb{Z}_p =\varprojlim \frac{\mathbb{Z}}{p^n\mathbb{Z}}$$ Then $\mathbb{Q}_p$, the field of the $p$-adic numbers ...
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Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units

On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem: Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\...
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Integer polynomials with roots in every $\mathbb{Z}_p$ but no rational roots.

I want to find polynomials in $\mathbb{Z}[x]$ with degree as small as possible such that these polynomials have no rational roots but have a root in the $p$-adic integers $\mathbb{Z}_p$ for every ...
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23 views

Embeddings of $K_v$ in $\mathbb{C}$

Let $K$ be a number field, $v$ a nonarchimedean prime, and $K_v$ the completion of $K$ at $v$. We have the embedding $K \to K_v$, and also $K \to \mathbb{C}$. I have two related questions: Is ...
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51 views

Isomorphism which involves $\mathbb Z_p[[T]] \otimes \mathbb Q_p$

Why should $\mathbb Z_p[[T]] \otimes_{\mathbb Z_p} \mathbb Q_p$ be isomorphic to the bounded sequences with values in $\mathbb Q_p$? The fact is that the tensor product is on $\mathbb Z_p$, so it is ...
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35 views

Codomain of p-adic logarithm

We have the natural map $$\log: \mathbb{C}^\times \to \mathbb{R}$$ $$z \to \log |z|$$ Is there a p-adic analogue of this? By this I mean, a map $\log_p: \mathbb{C}_p^\times \to \mathbb{Q}_p$, ...
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Metric on the profinite completion of the integers?

The p-adic integers come with a metric and associated topology, both of which can be restricted down to the integers. Does this also apply to the profinite completion of the integers? Do they have ...
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Formal power series over p-adic integers

Does $\mathbb Z_p \otimes \mathbb Z[[x]]=\mathbb Z_p[[x]]$ hold? In particular, I don't know how to express $\sum\frac 1{q^n}x^n$ for $q\ne p$ as an element of the tensor product. Or it should be $\...
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2answers
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Restriction from subgroup of the Galois group of max. unr, ext. $G(\tilde{K}/\mathbb{Q}_{p})$ to $G(K/\mathbb{Q}_{p})$ is surjective?

This is a question I'm struggling with for some time. Let $K$ be a finite Galois extension of $\mathbb{Q}_{p}$ and let $\tilde{K}$ denote the maximal unramified extension of $K$. We can then ...
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Introduction to p-adic vector spaces

I'm interested in learning about vector spaces over $\mathbb{C}_p$ and $\mathbb{Q}_p$. Most textbooks on p-adic numbers (Koblitz, Schikhof) focus on analysis and number theory. Is there any ...
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44 views

Upper Numbering of Ramification Groups of Absolute Galois Groups for Totally Ramified Extensions

Suppose $K'/K$ is a totally ramified extension of $p$-adic fields of degree $e.$ A paper (p.9, line 15) I am reading seems to use the following formula for the upper numbering on the absolute galois ...
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zeroes of homogeneous analytic $p$-adic functions

I am trying to understand Lemme 2.1 page 3 of this paper by Pilloni. What is says (I think) is that if you have, for a a positive real number $w$, an analytic function $$ f : \mathbf{Z}_p^\times(1+...
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$p$-adic Fourier transforms and orthogonality relations

In $\mathbb{C}$, we have the following orthogonality relation $$ \int_{0}^{1} e^{2\pi i (m-n)x} dx = \begin{cases} 1 & \mbox{ if } m = n;\\ 0 & \mbox{ otherwise.} \end{cases} $$ Do we have ...
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Solutions in $\mathbb Q_p$ leads to solution for congruences equations?

Let $p$ be a prime number such that $p\equiv1\pmod 3$. Let $n$ be an integer such that the equation $x^3=n$ has a solution in $\mathbb Q_p$. In fact with our assumptions, the others solution are in $\...
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1answer
38 views

Deducing Skolem-Mahler-Lech theorem from a $p$-adic interpolation result.

Let $p$ be a prime number, let $d$ be an integer greater than $1$ and let $f\in\mathbb{Z}_p[X_1,\cdots,X_d]^d$. I have already proved the following $p$-adic interpolation theorem: Theorem 1. (...
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A number $a$ is a square in $\mathbf{Q}$ if and only if it is a square in $\mathbf{R}$ and $\mathbf{Q}_p$ for all primes $p$

Problem from Schikhof's Ultrametric Calculus. As I understand it, the intersection of $\mathbf{R}$ and all $\mathbf{Q}_p$ is just $\mathbf{Q},$ so it seems that $x^2-a$ having a zero in $\mathbf{Q}$ ...
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Limit of a sequence in $\mathbb{Z}_p$ (J.-P. Serre, p-adic equations)

In a proof of a theorem in chapter 2 "p-adic equations" in "A Course in Arithmetic" from J-P Serre there is one conclusion that I don't understand. Here is the theorem I'm talking about (excluding the ...
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Axiomatic Approach to p-adic rationals

There are several ways to construct the real numbers: completion of the rationals w.r.t. the euclidean metric, dedekind completion of the rationals, infinite continued fractions, etc. Each of these ...
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Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
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Galois group action on etale cohomology groups

Let $X$ be a smooth and proper scheme over $Spec(\mathbb{Z}_p)$. Let $l$ be a prime number coprime to $p$. Then the proper base change theorem gives me an isomorphism $$H^r_{et}(X\times_{\mathbb{Z}_p}\...
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Embedding a number field in $\mathbb{Q}_p$.

Let $K/\mathbb{Q}$ be a finitely generated field extension and let $(x_1,\cdots,x_m)$ be a transcendance basis of $K/\mathbb{Q}$. Using primitive element theorem, there exists $y\in K$ algebraic over $...
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A degree comparison problem over p-adic fields

Consider multivariate polynomial $p(x)\in\Bbb Q_p(x)$ and multivariate rational function $r(x)\in\Bbb Q_p(x)$ where $x=(x_1,\dots,x_n)$. Assume that degree in each variable is $1$ but total degree ...
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An example of a discontinuous “$\ell$-adic Galois representation”

Let $\mathbb{F}_p$ be a finite filed with $p$ elements, and $G=\mathop{\mathrm{Gal}(\mathbb{F}_p^s/\mathbb{F}_p)}$ be its absolute Galois group. $G$ is a pro-finite group, with the Krull topology, see ...
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Isometries of p-adic vector spaces?

In Euclidean space $\mathbb{R}^d$ there are ${d+1 \choose 2}$ independent isometries (translations and rotations). In other words the dimension of the Euclidean isometry group is ${d+1 \choose 2}$. ...
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Understanding a proof of a corollary in chapter 2 about invertibility of a p-adic integer (Jean-Pierre Serre)

In a proof of a corollary in chapter 2, there is a step I don't understand. Corollary 2: Suppose $p \neq 2$. Let $f(X) = \sum_j a_{ij}X_iX_j$ with $a_{ij} = a_{ji}$ be a quadratic form with ...
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1answer
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P-adic expansion of rational

I want to find the 7adic expansion of 1/4. I found that this is …1515152, by using the algorithm of finding 1/4 = k + 7q for each digit. Is this correct?
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A question about the definition of $p$-adic pseudo-measure.

Let $\mathfrak B$ be a profinite abelian group and let $\Lambda(\mathfrak B)$ be defined as the inverse limit $\varprojlim \mathbb Z_p[\mathfrak B/ \mathcal H]$ where the inverse limit is taken with ...
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Changing uniformizer of $p$-adics

In the theory of $p$-adic fields typically a uniformizer $\pi$ is chosen that generates the maximal ideal, $m$. And a few theorems later it can be shown that every element $x \in O$ of the ring of ...
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Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
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II. p-adic equations [2.1. Solutions] (J.-P. Serre)

Currently, I'm reading a chapter about p-adic equations in A Course in Arithmetic by Jean-Pierre Serre and I have a hard time understanding it. My questions/thoughts are in textboxes like these. ...
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Limit of p-adic numbers

Let $\alpha\in\mathbb{Z}_p^\times$. I read somewhere that the limit of $\alpha^{n!}$ as $n\rightarrow\infty$ is equal to one. Can someone explain to me why this is? Thanks!
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Proving that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers)

I want to prove that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers). By the definition of inverse limit, we know that there is a ring homomorphism $\phi$ from $\mathbb Z_3$ to $\mathbb Z/...
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Rational sum of the $p$-adic series

Koblitz states (as one of the excercises to chapter 2) that whenever we are given an integer $k > 0$ and prime $p$, the series $$ f(p, k) = \sum_{n=0}^\infty n^kp^n $$ converges in $\mathbb Q_p$ ...
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Stronger form of Hensel's lemma?

Let $f \in \mathbb{Z}_p[x]$ and suppose $|f(a)|_p < |f'(a)|_p^2$ for some $a \in \mathbb{Z}_p$. Let $a_1 = a$, and for $n \ge 1$ let$$a_{n+1} = a_n - f(a_n)/f'(a_n).$$How do I see that this defines ...
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Subgroups of finite index of the split maximal torus of $GL_n(\mathbb{Z}_p)$.

Let $E_{i,j}$ be the $n \times n$ elementary matrices. Let $G=GL_n(\mathbb{Z}_p)$. Let $T_G$ be the split maximal torus of $GL_n(\mathbb{Z}_p)$. Let $\Theta$ be the subgroup of $T_G$ consisting of ...
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If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$?

If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$? I know we want to use Hensel's Lemma somehow to assess this question, but I'm ...
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Constructing Witt polynomials

I am reading about Witt vectors, and I keep seeing the following set of congruences often: For example, in these notes here, on page 3 we see the following congruences: \begin{align} \chi(c_0)^p+\chi(...
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P-adic expansions of Integers

Is there a way to prove that $a$ is an element of the Integers, if $p$ is prime, $a$ is in $\mathbb Z_p$ and its $p$-adic expansion is eventually periodic? I know how to do this for the rationals, ...
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Please express the first 3 7-adic digits of a root of $x^3-1$ in $\mathbb{Z}_7$ other than 1.

Please express the first 3 p-adic digits of a root of $x^3-1$ in $\mathbb{Z}_7$ other than 1. Does this just mean find the 7-adic expansion of 2 or 4? Wouldn't their expansions just be 2 and 4?
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$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
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Majoration of the $p$-adic valuation of a factorial.

Let $p$ be a prime number. In order to prove a result on $p$-adic interpolation of iterates, I need to show the following: Lemma. Let $m$ be an integer, one has: $$v_p(m!)\leqslant\frac{m}{p-1}....
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Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $

I read in a paper that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $ this is counterintuitive / surprising since $\mathrm{SO}_3(\mathbb{R}) \not \simeq \mathrm{SL}_2(\mathbb{R}) $ ...
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Can the zeroes of a multivariate $p$-adic polynomial be bounded?

Multivariate real polynomials, as opposed to multivariate complex polynomials, can have bounded zero sets, i.e. $x^2+y^2-1$ in $\mathbb{R}^2$. This fails in $\mathbb{C}^n$ because $\mathbb{C}$ is ...
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Find a matrix of order $p$ in a subgroup of $\operatorname{GL}_n(\mathbb Z_p)$

Let $p$ be a fixed prime and $\mathbb Z_p$ the ring of $p$-adic integers. Consider the subgroup $G_n\subseteq \operatorname{GL}_n(\mathbb Z_p)$ given by all matrices $(a_{ij})_{ij}$ such that $$ \...
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Serre mass formula under extension

Suppose $K_1$ is a local field and $K_2$ is a totally ramified finite Galois extension. Let $e$ be a positive integer with $[K_2:K_1]|e$. Consider the set of isomorphism classes of $K_1$ extensions of ...
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“Logarithmic derivative of a p-adic number”

Coleman, in "Division Values in Local Fields", (Inventiones math. 53, 91 - 116 (1979)), says that In his work on cyclotomic fields Kummer observed that various formal operations on power series ...
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infinitely $p$-divisible elements in $A\otimes \mathbb{Z}_p$

Let $A$ be a (possibly non-finitely generated) torsion-free abelian group. Suppose that $A$ contains no infinitely $p$-divisible elements, then does the same hold for $A\otimes \mathbb Z_p$, where $\...