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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How to calculate $\log_p x$ in p-adic analysis?

I'm studying p-adic analysis and recently I've learned about the p-adic logarithm function but I can't understand very well how the process of calculating the value should be done. As an exercise I'm ...
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Fast way to check if finite extension is unramified?

Consider the $5$-adic quadratic extension $\mathbb{Q}_{5}(\sqrt{5})/\mathbb{Q}_{5}$. I want to check if this extension is unramified, where unramified means that the corresponding extension of ...
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Solution of a equation in $\Bbb{Z}_p$

Let $m \in \Bbb{Z}_p$ be fixed. Let $a_1,...a_l$ be fixed integers. I am trying to find out solutions of the equation $m=x_1^{a_1}...x_l^{a_l}$ where $x_1,...,x_l\in \Bbb{Z}_p$. Here $x_1,...x_l$ are ...
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Quadratic extensions of p-adic rationals

In Alain Robert's A Course in p-adic Analysis, the author uses Hensel's Lemma to analyze quadratic extensions of $\mathbb{Q}_p$. He wants to calculate the index of $(\mathbb{Q}^*_p)^2$ in ...
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Prove that the sequence $(p^n)^{∞}_{n=1}$ for p a prime is a null sequence with respect to $| · |_p$.

Prove that the sequence $(p^n)^{∞}_{n=1}$ for p a prime is a null sequence with respect to $| · |_p$. Here is what I have: a null sequence is a sequence that maps to zero. $| · |_p=p^{-vp(\cdot)}$ ...
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A theorem on p-adic power series

This is a question about a proof of a theorem in many books on $p$-adic numbers and I don't seem to understand one of the directions. The theorem is Let $f(X) = \sum\limits_{n=0}^\infty a_nX^n \in ...
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Relation between Bombieri theorem and p-adic squares

Koblitz states in his book on p-adic numbers on page 84: Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all ...
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Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of ...
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Square of a p-adic number is an integer.

This is a question for an assignment, so just some pointers in the right direction would be great. Suppose that $n \in \mathbb Z$ and that $\alpha \in \mathbb Q_p$ satisfies $\alpha^2 = n$. Prove ...
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Understanding proof that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is unramified for $(n,p)=1$.

Problem Consider the extension $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$, where $\zeta$ is a $n$-th primitive root of unity and $(n,p)=1$. I want to show that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is ...
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n-th power residues in p-adic fields

How to see that number of n-th power residue classes of a p-adic field K (elements of $K^*/K^{*n}$) is finite? I know how to prove using Hensel lemma that all p-adic integers sufficiently close to 1 ...
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Prove that $\mathbb{Z}_p = {a\in\mathbb{Q}_p:|a|_p\leq 1}$.

Let $\mathbb{Z}_p= {a\in\mathbb{Q}_p : a=\sum_{i=0}^{\infty}d_ip^{i},0\leq d_i\leq p-1\;\forall\;i\geq 1}$ be a subset of $\mathbb{Q}_p.$ In other words, $\mathbb{Z}_p$ is the set of all p-adic ...
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Minimal polynomial has coefficients in the valuation ring?

Question Let $K$ be a complete nonarchimedean valued field with valuation ring $\mathcal{O}$ and let $L/K$ be a finite extension. Let $\alpha$ be an element of $L$ and $f\in K[x]$ its minimal ...
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Cuspidal and Supercuspidal representation

Let $G$ be an algebraic group over a field $F$, and let $(\pi,V)$ be a smooth $G$-representation over an algebraic closed field $k$. Then $\pi$ is called a CUSPIDAL representation if $r(V)=0$ for any ...
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Let $p_1,p_2\in\mathbb{Z}$ be distinct prime numbers. Show that $|\cdot|_{p_1}\not\sim|\cdot|_{p_2}$

Let $p_1,p_2\in\mathbb{Z}$ be distinct prime numbers. Show that $|\cdot|_{p_1}\not\sim|\cdot|_{p_2}$ by finding a sequence in $\mathbb{Q}$ that is Cauchy with respect to $|\cdot|_{p_1}$ but not Cauchy ...
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Prove that the sequence $(s_n)$ in $\mathbb{Q}$,

Prove that the sequence $(s_n)$ in $\mathbb{Q}$, where $$\sum_{i=-m}^{n}d_i,p^i$$ with $0\leq d_i\leq p-1$ and $m\geq0$ represents a p-adic number. Here is what I have so far: Set $\epsilon>0$ ...
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Showing the sequence of partial sums of a power series is Cauchy under the p-adic norm

I want to show that the sequence $e_n = \sum_{i = 0} \frac{x^i}{i!}$ is Cauchy with respect to the p-adic norm when $p > 2$ and $|x|_p < 1$. So far, I have the estimate $|e_n - e_m|_p = |\sum_{i ...
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'Smooth' p-adic analysis (perhaps via toposes)

There are sensible theories of analytic functions on non-Archimedean fields (rigid analytic spaces, Berkovich spaces), but these are modeled after complex analysis. I'm curious to what extent there ...
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Prove that $a \equiv b \mod{p^j} \iff |a-b|_{p}\leq p^{-j}$

Prove that $a \equiv b \mod{p^j} \iff |a-b|_{p}\leq p^{-j}$ Typically, when dealing with a congruence I go to the division statement. i.e $$a\equiv b\mod{p^j}\Rightarrow p^j|a-b \;\;\;(\star)$$ ...
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Reconstructing formal groups from the p-map, realizing p-maps from formal group

Suppose $F$ is a formal group over $\mathbb{Z}_p$. There are few trivial condition that $f \in \mathbb{Z}_p[[t]]$, the power series representing the $p$-map should satisfy: 1)$f \equiv g(t^p) mod p$ ...
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Integral closure of the p-adic integers in a finite extension of the p-adic numbers

In Cassels' article "Global Fields", he uses the term "ring of integers" and the notation $\mathcal{O}_K$, where $K$ is a field with a non-archimedean valuation, to denote the ring of elements $x \in ...
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Quotient of $\mathbb{Z}_p$ by the rational integers

Let $p$ be a prime and let $\mathbb{Z}_p$ denote the $p$-adic integers. One has a canonical inclusion of rings $$\mathbb{Z}_{(p)}\longrightarrow\mathbb{Z}_p$$ given by identifying the rational ...
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Showing that $ord_p((p^n)!) = 1 + p + \dots + p^{n-1}$

I need to prove that $ord_p((p^n)!) = 1 + p + \dots + p^{n-1}$ ($ord_p$ is the $p-$adic valuation), which is essentially a nasty combinatorics problem. I want to count how many integers from $1$ to ...
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50 views

p-adic density of zeroes of a polynomial

I saw the following definition of the p-adic densities of zeroes of a system of polynomial equations: Definition: Suppose that we have a system of homogeneous polynomials of degree $d$, $ ...
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Valuations above $\mathbb{Q}(\alpha)$

Let $K$ be a field, we say that a function $\nu:K\to \mathbb{Z}_{\geq 0} \cup \lbrace \infty \rbrace$ is a valuation above $K$ if then followings hold for each choose of $x,y \in K$: 1 $\nu(x) = ...
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Closed subgroups of $Z_{p}^{\times}$

I was able to prove that any closed subgroup of additive group of $Z_{p}$ is of the form $p^{n}Z_{p}$ for some $n$. I asked the same question for the multiplicative group of units in $Z_{p}$, that is ...
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What is a “branch” of p-adic exponentiation?

I am reading p-adic Numbers, p-adic Analysis, and Zeta-Functions by Neal Koblitz. Please look at page 27. The definition of $f(x)=n^x$ is unambiguously defined when $n$ is $1 \mod p$. Next, for any ...
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Examples of Cauchy sequences in the rational numbers that do not converge in said set with respect to the p-adic topology

I am working through a set of notes on Algebraic Number theory, and I find myself attempting to construct some examples of Cauchy sequences not converging in $\mathbb{Q}$, with respect to the p-adic ...
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Density of the image of the set $\lbrace (x,x), x\in \mathbb{Z} \rbrace $ in $\mathbb{Z_{p}} \times \mathbb{Z}_{q}$

If $p,q$ are distinct primes, it is true that the subset $\mathbb{Z} \times \mathbb{Z}$ is dense in $\mathbb{Z}_{p} \times \mathbb{Z}_{q}$. However, is it true that $\lbrace (x,x), x\in \mathbb{Z} ...
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about norm and p - adic number

For $x \in {\mathbb{C}_{p}}$ and $x \neq 1$ and $x^{p}=1$ with $p$ is a prime number. Then can we calculate $ {\left |x-1 \right |}_{p} ?$ We knew that ${\left | .\right |}_p$ is p -adic norm on ...
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Metric on $\mathbb{Q}$ for which the completion of $\mathbb{Q}$ is isomorphic to $\mathbb{Q}_2 \times \mathbb{Q}_3$. [closed]

Is there a metric on $\mathbb{Q}$ for which the completion of $\mathbb{Q}$ is isomorphic to $\mathbb{Q}_2 \times \mathbb{Q}_3$?
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Creating a grid of coloured points

I wish to create a grid of $31\times31$ points with coordinates of the form $(\frac k{30};\frac n{30})$ within the unit square, and give each point a colour based on the 2-adic value of both its ...
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70 views

Structure of $\Bbb Q_p/\Bbb Q$

The group $\Bbb A_\Bbb Q/\Bbb Q$ of adeles mod rationals is: Isomorphic to the solenoid Isomorphic to the group $\Bbb A_\Bbb Z/\Bbb Z$ Dual to the rational numbers This raises the question of the ...
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Error when tensoring with $p$-adic integers

It must be simple but I cannot find the error in the reasoning. Let $p$ be a prime number, we know that $\mathbb Z_p$ is flat over $\mathbb Z$ so that we can take the tensor product of the exact ...
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“Prime decomposition” for the profinite integers, adeles, or p-adics?

The strictly positive integers can be decomposed into a product of prime powers. Likewise, the positive rationals can be decomposed into a product of (possibly negative) prime powers. Another way to ...
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Is a locally constant function from a profinite abelian p-group constant modulo an open normal subgroup?

Suppose $f$ is a p-adic valued function from a profinite abelian group $B$ ($Z_{p}$ or $Z_{p}^{*}$) such that for all $x$ in $B$ there is an open set $N(x)$ around $x$ such that $f$ is constant on ...
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Exactly 50% of 2-adic integers have the norm equal to $1$, exactly 25% have the norm $\frac{1}{2}$ and so on?

The question itself is more general and relates to all p-adic numbers, but it's really easy to show the principle using 2-adics. The definition of p-adic norm in most textbooks is not easy to ...
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Intuition behind Hensel's Lemma

In an Elliptic Curves course, my lecturer states Hensel's Lemma as the following: Let $k$ is a field that is complete with respect to a non-archimedian norm $|.|$ and $$R=\{x \in k : |x| \leq ...
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How does one visualize elements of standard Iwasawa algebra?

How does one think of elements of standard Iwasawa algebra of a profinite abelian group $B$ ? I mean like elements of $Z_{p}$ are represented as infinite series, is there a way to think of elements of ...
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Is $1+T$ a topological generator for $Z_{p}[[T]]$?

Is $1+T$ a topological generator for $Z_{p}[[T]]$? ($Z_p$ is the ring of p-adic integers)
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equivalent norms on vector space

For $(F,\left \| . \right \|)$ is a field with local compact norm and $V$ is a vector space over F. Can we have all norms on $V$ are equivalent?
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Fundamental unit of real quadratic field

Let $K=\mathbb{Q}(\sqrt d)$, where $d$ is squarefree and greater than $1$. Assume that $5|(d+1)$ or $5|(d-1)$. Let $\mathfrak{P}$ be a prime of $K$ over $5$. Let $\delta _o$ be a fundamental unit of ...
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P-adic numbers and infinity? Does infinity as a limit exist for p-adics?

I don't think I understand how p-adic numbers relate to the usual concept of infinity. The wiki page and various sources on the internet did not help. Let's see for example the 10-adic counterparts ...
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Distinct zeros of polynomials in $5$-adics.

How many distinct zeros does each of the following polynomials have in $\mathbb{Z}_5$? $f(x) = x^3 + 5x + 5$; $g(x) = x^5 + 2$; $h(x, y) = x^2 + y^2$. I know how to do the first ...
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zeros of p-adic power series

Suppose I have $f(x) \neq 0 \in \mathbb{Z}_p[[x]].$ If it can help it is of the form $g(x^p)+ph(x)$. Under which conditions can I say that $f(x)$ has finitely many zeros in $\bar{\mathbb{Q}}_p$? ...
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Understanding $Gal(\bar k /k)$

According to this book, An Introduction to the Langlands Program, One of the fundamental goals of modern number theory is to understand the Galois group $Gal(\bar k /k)$ where $k$ is a local or a ...
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Condition inverse $p$-adic number

Take $p$ prime, $n \in \mathbb{Z}_{>0}$ and $x \in \mathbb{Z}_p$. Suppose that $p$ isn't a divisor of $$x = (x_j + p^j \mathbb{Z})_{j \in \mathbb{Z}_{>0}},$$ then one can prove that the first ...
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$\mathbb{Z}_p$ is a complete metric space in which $\mathbb{Z}$ is dense

I don't quite understand a few parts of the proof of proposition $3$. What is meant by "the ideals $p^n\mathbb{Z}_p$ form a basis of neighborhoods of $0$"? After reading the definition of a ...
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49 views

Let $p$ and $q$ be distinct primes. Can you prove the sequence $\{p^n\}_{n \in \mathbb{N}}$ is not Cauchy under the given metric on $\mathbb{Q}$?

This is an elementary $p$-adic theory question. Granted $d(x,y)=|x-y|_q$ is a metric on $\mathbb{Q}$, and $|\cdot|_q$ is a norm such that $$|x|_q=q^{-ord_q x}$$ where $ord_q x$ is the largest ...
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81 views

profinite completion of a ring of integers in a global field

It is well known that the profinite completion of the integers, $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z}$ is, by the Chinese remainder theorem, isomorphic to $\prod_p \mathbb{Z}_p$. I ...