Questions tagged [p-adic-number-theory]

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.

This tag is for $p$-adic number systems. Read more in this Wikipedia article.

1994 questions
14
votes
1 answer

Units of p-adic integers

How to prove that the $p$-adic units can be written as $$\mathbb{Z}_p^\times \cong \mu_{p-1}\times(1 + p\mathbb{Z}_p) \cong \mathbb{Z}/(p-1)\mathbb{Z}\times\mathbb{Z}_p$$ where $\mu_n$ is the $n$-th roots of unity in $\mathbb{Z}_p$? Here $p>2$ is a…
Akatsuki
  • 3,050
  • 1
  • 15
  • 32
14
votes
3 answers

Method of finding a p-adic expansion to a rational number

Could someone go though the method of finding a p-adic expansion of say $-\frac{1}{6}$ in $\mathbb{Z}_7?$
user84899
  • 445
  • 6
  • 12
13
votes
1 answer

Is $\mathbb Q_r$ algebraically isomorphic to $\mathbb Q_s$ while r and s denote different primes?

It is obvious that $\mathbb{Q}_r$ is topologically isomorphic to $\mathbb Q_s$ while $r$ and $s$ denote different primes. But I really don't know whether it is true in the aspect of algebra. As I failed to prove it, I think that it is false, but I…
gottigen
  • 205
  • 1
  • 6
13
votes
3 answers

Tensor product of a number field $K$ and the $p$-adic integers

In the paper A database of local fields, J. Jones and D. Roberts introduced an isomorphism $K \otimes \mathbb{Q}_p \cong \prod\limits_{i=1}^g K_{p,i}$, where $K$ is some finite dimensional extension of $\mathbb{Q}$ and $\{K_{p,i} \}$ is some…
D_S
  • 30,734
  • 6
  • 38
  • 107
13
votes
0 answers

Is there a proof of quadratic reciprocity using $p$-adic numbers?

I know that the quadratic reciprocity can be regarded as a special case of Artin reciprocity (class field theory), and we can get it by considering the cyclotomic extension of $\mathbb{Q}_{p}$. However, I want to know if there's any proof of…
13
votes
2 answers

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}) $ I don't know about $p$-adic numbers, but for…
cactus314
  • 23,583
  • 4
  • 35
  • 88
12
votes
2 answers

Why are closed balls in the $p$-adic topology compact?

I was skimming through some of this paper Measurable Dynamics and Simple $p$-adic Polynomials out of curiosity. A few pages in, the author claims that closed balls are both open and compact sets in the $p$-adic topology on $\mathbb{Q}_p$. I have…
Buble
  • 1,509
  • 12
  • 16
12
votes
2 answers

The p-adic numbers as an ordered group

So I understand that there is no order on the field of $p$-adic numbers $\mathbb{Q}_p$ that makes it into an ordered field (i.e.) compatible with both addition and multiplication. Now, from the responses to a couple of my previous questions,…
B M
  • 920
  • 4
  • 17
12
votes
2 answers

A puzzle involving $10$-adic numbers

If one iterates the squaring: $$5^2 = 25, 25^2 = 625, 625^2 = 390625 $$ one quickly notices that the end digits become 'stable', i.e. however far you take the above procedure, the result will always end in $...90625.$ The resulting 'limit' is a…
12
votes
5 answers

Proving $\sqrt{2}\in\mathbb{Q_7}$?

Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$? I understand Hensel's lemma, namely: Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers such that $m \leq k$. If $r$ is an integer such that…
12
votes
2 answers

Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$

The following is a historical question, but first some background: Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be shown to be equal (even for an infinite…
12
votes
2 answers

Is there a univariate rational polynomial which represents only squares in $\mathbb{R}$ and $\mathbb{Q}_2$, but not all other $\mathbb{Q}_p$?

Let $K$ be a field; I will say a polynomial $f \in K[X]$ represents an element $a \in K$ if there exists a $b \in K$ such that $f(b) = a$. Denote by $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{Q}_p$ the fields of rational, real and $p$-adic numbers…
Bib-lost
  • 3,730
  • 11
  • 36
12
votes
2 answers

Are the p-adic integers the ring of integers of the field of p-adic numbers?

This question was much simpler, but as I was typing it, it became a chain of questions. My starting question was Is $\mathbb{Z}_p$ (obtained by the inverse limit procedure with the directed system $\cdots \to \mathbb{Z}/p^2\mathbb{Z} \to…
RKD
  • 8,075
  • 22
  • 45
12
votes
1 answer

Why do we define the $\mathfrak{p}$-adic logarithm on a $\mathfrak{p}$-adic number field such that $\log(p) = 0$?

Suppose we have a finite extension $K / \mathbb{Q}_p$ with valuation ring $\mathcal{O}$ and maximal ideal $\mathfrak{p}$. One can define the $\mathfrak{p}$-adic logarithm on the group of principal units $U^{(1)}$ of the local field $K$ using the…
11
votes
2 answers

Open problems involving p-adic numbers

I am in my final year of my undergraduate degree, and I'm doing a project on p-adic numbers, and in particular, trying to find Galois groups of simple extensions of $\mathbb{Q}_p$ (this is a Galois theory course). I have developed and become…