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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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### Connection between roots in $Q_p$ and $Z_p$

I'm working on some problems where I have to find solutions in $Z_p$ and $Q_p$ of polynomials of the form $ax^2+by^2=1$. I've seen Hensel's lemma for solution over $Z_p$. For solutions over $Q_p$. I'...
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### How to prove the p-adic expansion of negative p-adic integers ends with p-1?

I just started calculating with p-adic numbers and one of the questions I'm stuck on is how I can prove that for a negative integer the p-adic expansion ends with an infinite amount of digits p-1. I ...
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### On the concept of primary element

Let $\ell$ be an odd prime number,$\zeta_{\ell}:=e^{2\pi i/\ell}$, $F:=\mathbb{Q}(\zeta_{\ell})$ be a cyclotomic field, $\mathcal{O}_F$ be its integer ring and $\lambda:=1-\zeta_{\ell}$. [Ireland-...
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### What is the decomposition of global units $1+\mathfrak{p}$?

Let $p \geq 2$ be prime and $K=\mathbb Q(\zeta_p),~\zeta^{p}=1$ with ring of integers $\mathcal{O}_K$. we denote by $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ be ...
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### Why's $v_L (x)=\frac{1}{n}v_K(N_{L/K}(x))$ integer for unramified local field extension $L/K$?

If $K$ is a local field and $L/K$ is a finite extension, then the valuation $v_K$ can be extended uniquely to a valuation $v_L$ of $L$ such that $v_L$ restricted to $K$ is equal to $v_K$. This is one ...
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### $\mathbb{Z}_p$ is a compact subgroup

Prove that $\mathbb{Z}_p \subset \mathbb{Q}_p$ is a compact subgroup and show that this is the maximal compact subring. Hint: use that every sequence of integers had a Cauchy sub-sequence with ...
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### On the Iwasawa Algebra

I am reading Joaquin Rodrigues Jacinto's and Chris Williams' notes on $p$-adic $L$-functions http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/Number-Theory---Full-Lecture-Notes-2017-...
1 vote
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### Classification of branched division maps on $\mathbb R/\mathbb Z$?

Consider the doubling map $x\mapsto 2x$ on $\mathbb R/\mathbb Z$. I'm interested in the collection of all right inverses of this map whose image is an interval. In other words, I want to know about &...
1 vote
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### Legendre symbol on $p$-adic integers $\mathbb{Z}_p$

I have seen that you can define the usual modular arithmetic on the $p$-adic integers: For $a,b\in\mathbb{Z}_p$ and a prime $p$, $$a\equiv b\pmod{p}\iff (a-b)/p\in \mathbb{Z}_p.$$ My question is, can ...
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### What functions on 2-adic completion of the rationals are in the conjugacy class of $x+2^{\nu_2(x)}$?

Question What functions on 2-adic space are in the conjugacy class of $f(x)=x+2^{\nu_2(x)}$? A conjugacy class is the set of functions whose action conjugates to any other element of the class, in ...
96 views

### Let $K$ be a number field and $v$ be its place. Let $K_v$ be completion of $K$ at $v$. I want to prove $y^2=x^4-p$ has $K_v$ rational point.

Let $K$ be a number field and $v$ be its place. Let $p$ be a prime element of ring of integers of $K$. Let $K_v$ be completion of $K$ at $v$. I want to prove $C: y^2=x^4-p$ has $K_v$ rational point. I ...
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### "p-adic completion" of rational function field

Let $q$ be a prime power, $\mathbb{F}_q$ the finite field of order $q$ and let $\mathbb{F}_q(T)$ be the field of rational functions over $\mathbb{F}_q$. Similarly to $\mathbb{Q}$, the absolute values ...
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### Showing an abelian pro-$p$ group is a $\mathbb{Z}_p$ module

I am rather new to $p$-adics, and I am trying to show that every abelian pro-$p$ group is a $\mathbb{Z}_p$ module. I know some partial answers already exist (About pro-$p$ groups), but I am having ...
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### What is the 2-adic expansion of 1/2?

I understand that -1/3 can be represented in 2-adics as $01010101...$, and 1/3 is just $1101010101...$. What about 1/2, though? Is it possible to represent it as an infinite expansion, or is the only ...
1 vote
87 views

### Similarity of Lifting the Exponent Lemma in Pell numbers

Pell number is a term of the sequence $\{P_n\}$ determined by a recurrence relation $$P_{n+2}=2P_{n+1}+P_n, P_0=0, P_1=1.$$ Let $v_p(x)$ be the $p$-adic valuation of an integer $x$ (the number of ...
62 views

In this note from Keith Conrad, he explains an interesting application of Strassmann's theorem to the divergence of certain linear recurrence integer sequence. More precisely, the sequence defined as $... 0 votes 0 answers 62 views ### Is$\Bbb{Q}_p/\Bbb{Z}_p$isomorphic to$\Bbb{Q}/\Bbb{Z}$as aan abelian group? Is$\Bbb{Q}_p/\Bbb{Z}_p$isomorphic to$\Bbb{Q}/\Bbb{Z}$as a group ? My try: I know$\Bbb{Q}/\Bbb{Z}\cong \bigoplus_p:prime　\Bbb{Q}_p/\Bbb{Z}_p$as group. If I could prove$\bigoplus_p:prime　\Bbb{Q}...
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Let $f(x)=\displaystyle{\sum_n a_nx^n}$ be a power series with coefficients in the field of p-adic rationals, $\mathbb{Q}_p$. Let $R_f$ be its radius of convergence and let $f'$ denote its term-by-...
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### Take limit in p-adic integer ring: the difference between $\lim_{n \rightarrow \infty} \alpha^{n!}$ and $\lim_{n \rightarrow \infty} \alpha^{n}$

Let $p$ be a prime, $L / \mathbb{Q}_p$ be a finite field extension. Let $\mathcal{O}_L$ its valuation ring (ring of local integers) in $L$, $\mathfrak{m}_L$ be the unique maximal ideal, and the ...
### Is $\log_p(x^z)=z \log_p(x)$ for $z \in \mathbb Z_p$?
Let $\log_p$ denote the $p$-adic logarithm and $\mathbb Z_p$ be the ring of integers. We know that $p$-adic logarithm has no base, but still we have $\log_p(x^n)=n \log_p(x)$ for $n \in \mathbb Z$ and ...
### $f(T)=T^2 + 1$ irreducibility over $p$-adic number polynomials, $\mathbb Q_p[T]$.
In relation to this previous post, Why is $x^2+1$ irreducible in $\mathbb{Q}_2[x]$., is there something we can say about $f(T) = T^2+1$ in generality over different choices of $p$ of $\mathbb Q_p[T]$? ...