# Tagged Questions

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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### 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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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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### 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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### Do the number of degree p extensions of p-adic fields lie in a recursive sequence? And if so, why?

I noticed something on this page, that may just be coincidental: http://www.lmfdb.org/LocalNumberField/ From inspecting the table there, you can conclude that most of the interesting extensions of ...
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### Completions of number fields

I would like to prove a statement about completions of number fields, but I'm running into a problem. The statement I want to prove is Let $L/K$ be a Galois extension of number fields, $p$ a prime ...
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### What does extra zero of an $L$-function mean?

This is a very vague question. What does an extra zero of an $L$-function mean? There are lots of papers written on this topic, investigating the extra/exceptional zeros of various $p$-adic $L$-...
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### Finite extensions of $\mathbb Q_p$ are exactly completions of numberfields

I read that every finite extension of $\mathbb{Q}_p$ is in fact a completion of a numberfield K with a place of K. I also heard that this is a consequence of Krasner´s Lemma. Do you have any hint how ...
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### What constitutes a good reading course in $p$-adic number theory?

I have had a course in number theory where I studied Marcus and also a course in differential geometry. I have read Koblitz's introductory book on $p$-adic numbers. I am roughly interested in both $p$-...
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### Find the first six terms in a reduced coherent sequence defining $i$ in $\mathbb{Z}_5$.

Find the first six terms in a reduced coherent sequence defining $i$ in $\mathbb{Z}_5$. I need to use the sequence $a_k=2^{5^{k-1}}$ but not sure how to? Any hints?
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### Congruence subgroup of $\mathbb{GL}_n(\mathbb{Z}_p)$

In course of my research I met the following situation : 1) I have a bunch of open subgroup (so of finite index) in $\mathbb{GL}_{n}(\mathbb{Z}_p)$. 2) My groups arises naturally as stabilizers of ...
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### $4$th root of unity: 5-adic

For $k \geq 1$, let $x_k = a^{p^{k-1}}$. Taking $p = 5$ and $a = 2$, find the first six terms in a reduced coherent sequence defining a $4$th root of unity (i.e. $\sqrt{−1}$) in $\mathbb{Z_5}$, and ...
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### $\mathrm{ord}_p(x)$ and convergence in $\mathbb{Q}_p$

Let $x=\frac{22}{7} \in \mathbb{Q}$. (a) Find $\mathrm{ord}_p(x)$ for all primes $p$. $\mathrm{ord}_2(x)=1,\ \mathrm{ord}_{11}(x)=1,\ \mathrm{ord}_7(x)=-1$ and $\mathrm{ord}_p(x)=0$ for all ...
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### $5$-adic expansion

Given $x = 2+1\cdot5+3\cdot5^2 +2\cdot5^3 +\ldots ∈ \mathbb{Z_5}$, find $\frac{1}{x}$, expressing it similarly as a $5$-adic expansion. (First 4 digits only). I'm new to p-adic numbers and was ...
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### Kummer map and cohomology group for an elliptic curve

Let $E=E_q$ be the Tate ellipitc curve over a finite extension $K$ of $\mathbb{Q}_p$ for a $q$. Let $T$ be its p-adic Tate module. Let $\mathfrak m$ be the maximal ideal in $K$. I saw in this paper (...
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### Equality of $p$-adic fields

How can I prove that $\mathbb{Q}_3(\sqrt{2})=\mathbb{Q}_3(\sqrt{5})$. I only did modulo reduction, but how to prove it directly?
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### Connected Components of p-adic rationals

Notation: $p$ - a prime integer, $\Bbb{Z}_p$ - set of $p$-adic integers, $\Bbb{Q}_p$ - set of $p$-adic rationals, $\Bbb{Q}$ - set of rationals, $\Bbb{R}$ - set of reals. While reading up on $p$-...
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### Is I-adic completion a ring epimorphism?

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about ...
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### How algebraically restrictive is the structure of $\Bbb Q_p$?

My question is a little difficult to explain, but I will try to state it. First, let me talk about the field $\Bbb R$ of real numbers. Let $K$ be the maximal real algebraic extension of $\Bbb Q$, that ...
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I am trying to figure out the valuation of n! with respect to some p-adic norm p. This is part of a proof about the convergence behavior of $x^n /n!$ in the p-adics. When I try and expand out n into ...
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### examples of unramified extensions of $\mathbb{Q}_p$

For every local field $K$ and natural number $n$ coprime to $K$'s residue characteristic, there is a unique unramified extension $L/K$ of degree $n$. Let's take $K=\mathbb{Q}_p$. What are some ...
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### A fundamental tool used in the study of Diophantine equations…

Notations: $K$ is a perfect field, $\overline K$ an algebraic closure and $V\subseteq \mathbb P^n(\overline K)$ is a projective variety on $\overline K$. If $V$ is defined over $K$, in symbols $V/K$,...
### Are the $p^n$-adic numbers isomorphic to the $p$-adic numbers?
I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the ...