The p-adic-number-theory tag has no wiki summary.
17
votes
3answers
977 views
How far are the $p$-adic numbers from being algebraically closed?
A few days ago I was recalling some facts about the $p$-adic numbers, for example the fact that the $p$-adic metric is an ultrametric implies very strongly that there is no order on $\mathbb{Q}_p$, as ...
11
votes
2answers
244 views
Why is $O_K\otimes \mathbb{Z}_p\cong \oplus_{\mathfrak{p}|p}O_{K,\mathfrak{p}}$?
In my old number theory notebook this is stated as a fact. However, I ran into problems when I tried to prove it. First let me state the (supposed) theorem accurately:
Theorem (?)
Let $K$ be a ...
10
votes
1answer
166 views
Are all $p$-adic number systems the same?
After just having learned about $p$-adic numbers I've now got another question which I can't figure out from the Wikipedia page.
As far as I understand, the $p$-adic numbers are basically completing ...
10
votes
1answer
84 views
Show that $x^4+1$ is reducible in p-adic numbers $\mathbb{Q}_p$ for p>2 prime.
This is a homework problem for algebraic number theory but I'm having trouble getting started. Do I use induction in general, or show this holds for $p \equiv 1,3$ (mod 4)?
Any help would be ...
9
votes
4answers
265 views
Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?
I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this:
Show that $x^2-82y^2=\pm2$ has solutions in every ...
9
votes
1answer
108 views
diagonalizing a matrix over the $\ell$-adics
Let $M$ be a $2 \times 2$ matrix with coefficients in $\mathbb{Z}_{\ell}$ whose characteristical polynomial is
$$
P(T) = T^2- (a+d) T + (ad-bc).
$$
I've encountered the following assertion: If ...
9
votes
1answer
136 views
Local solutions of a Diophantine equation
I am trying to prove that the equation
$$3x^3 + 4y^3 +5z^3 \equiv 0 \pmod{p}$$
has a non-trivial solution for all primes $p$.
I am sure that this is a standard exercise, and I have done the easy ...
8
votes
2answers
438 views
Are all nonarchimedean valuations discrete?
I am studying valuation theory on the way to local class field theory, and the texts I have looked at immediately focus on discrete valuations in developing the theory of nonarchimedean valuations. ...
8
votes
2answers
557 views
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 ...
8
votes
1answer
168 views
What is the $p$-adic zeta function?
I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in ...
7
votes
2answers
283 views
Roots of unity in $\mathbb{Q} _{11}$
Here $\mathbb{Q} _{11}$ denotes the 11-adic field.
How can I show that the only root of unity of order 7 in this field is 1?
Is it true that for any two distinct primes $p,q$, the only root of unity ...
7
votes
1answer
88 views
Set of locations where the Hilbert symbol is not equal to $1$
Let $V$ be the set of prime together with the symbol $\infty$. For a prime $v=p$, denote the $p$-adic numbers by $\mathbb{Q}_p$ and the real numbers by $\mathbb{Q}_\infty$. For $v\in V$ the Hilbert ...
6
votes
2answers
324 views
Is the algebraic closure of a $p$-adic field complete
Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?)
Why is (or why isn't) an algebraic closure $\overline{K}$ complete?
Maybe this holds more ...
6
votes
2answers
654 views
What do the $p$-adic roots of unity look like?
I know that $\mathbb{Z}_p$ has all the $p-1^{st}$ roots of unity (and only those). Is it true that mod $p$ they are all different? Meaning, is the natural map $\mathbb{Z}_p \rightarrow \mathbb{F}_p$, ...
6
votes
2answers
352 views
$n$th powers in the p-adics
Suppose $K$ is a $p$-adic field (finite extension of the $p$-adics), and let $n$ be any integer (independent of what $p$ is). Define $U$ to be the set of all $x$ in $K$ such that $|x| = 1$ and such ...
6
votes
2answers
510 views
Binomial coefficients: how to prove an inequality on the $p$-adic valuation?
In section 4 of the article by Afred van der Poorten's A Proof That Euler Missed ... the following inequality is used:
$$\nu_{p}\displaystyle\binom{n}{m}\leq\left\lfloor\dfrac{\ln n}{\ln ...
6
votes
2answers
146 views
Does $\mathbb{Q}_p \cap \overline{\mathbb{Q}}$ depend on the embedding of $\overline{\mathbb{Q}}$ into $\overline{\mathbb{Q}_p}$?
I think the title pretty much says it all. I'm getting confused in the subtle parts of a proof, and would appreciate some help.
6
votes
2answers
151 views
Easy way to determine the primes for which $3$ is a cube in $\mathbb{Q}_p$?
This is a qual problem from Princeton's website and I'm wondering if there's an easy way to solve it:
For which $p$ is $3$ a cube root in $\mathbb{Q}_p$?
The case $p=3$ for which $X^3-3$ is not ...
6
votes
3answers
206 views
A property of some sequences of natural numbers (and their binary representation)
Let's consider a sequence of natural numbers $a_n$, represented in binary, with the following properties:
$\forall n \in \mathbb{N}$ the number $a_n$ is represented with $n$ binary digits
$\forall ...
6
votes
1answer
67 views
Does there exist an ordered ring, with $\mathbb{Z}$ as an ordered subring, such that some ring of p-adic integers can be formed as a quotient ring?
I'm not sure if this might instead be MathOverflow material, but I'll give it a shot here first anyway.
Does any ring $R$ exist that satisfies the following properties?
$R$ is a totally ordered, ...
6
votes
2answers
236 views
What does the Haar measure on $\hat{\mathbf{Z}}$ look like?
What does the Haar measure on $\hat{\mathbf{Z}} = \prod_p \mathbf{Z}_p$ look like?
Does it bear any relation to the "upper density" of a subset $S\subset\mathbf{N}$ defined by $m(S) = ...
6
votes
1answer
177 views
Line in a proof on p69 in Cassel's Local Fields
I'm trying to read the proof of
LEMMA 6.1 (Nagell)
Let $u_n$ be defined by $u_0=0$, $u_1=1$ and
$u_n=u_{n-1}-2u_{n-2} \hspace{20pt} (n\geq 2)$.
Then $u_n=\pm1$ only for $n=1,2,3, ...
6
votes
1answer
161 views
The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}$.
The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}=\{\sum_{i=0}^\infty a_ip^i:0\le a_i \le p-1\}$.
I was trying ...
6
votes
1answer
118 views
$\mathbb Z_p$-rank $\leq r_1+r_2-1$ in Leopoldt's conjecture
I am trying to understand Leopoldt's conjecture as formulated in section 5.5 of Washington's "Introduction to Cyclotomic Fields".
For an algebraic number field $K$ let $E$ denote the global units, ...
6
votes
1answer
178 views
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 ...
5
votes
2answers
197 views
Question about $p$-adic numbers and $p$-adic integers
I've been trying to understand what $p$-adic numbers and $p$-adic integers are today. Can you tell me if I have it right? Thanks.
Let $p$ be a prime. Then we define the ring of $p$-adic integers to ...
5
votes
5answers
176 views
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 ...
5
votes
3answers
198 views
Finding $p^\textrm{th}$ roots in $\mathbb{Q}_p$?
So assume we are given some $a\in\mathbb{Z}_p^\times$ and we want to figure out if $X^p-a$ has a root in $\mathbb{Q}_p$. We know that such a root must be unique, because given two such roots ...
5
votes
1answer
100 views
Why $p$-adically interpolate?
I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
5
votes
1answer
69 views
equation over $\mathbb{Z}_3$
Consider the ring $\mathbb{Z}_3$ of 3-adic integers. Does there exist a positive integer $n$ and a solution to $(X_1^2 + X_2^2 + \cdots + X_{n - 1}^2)^2 = 2X_n^4$ in $\mathbb{Z}_3^n$? If so, what is ...
5
votes
1answer
194 views
Integral closure of p-adic integers in maximal unramified extension
Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
5
votes
1answer
84 views
Tensor products of p-adic integers
These are relatively simple questions, but I can't seem to find anything on tensor products and p-adic integers/numbers anywhere, so I thought I'd ask.
My first question is: given some ...
5
votes
2answers
455 views
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 ...
5
votes
1answer
57 views
p-adic expansion
Let $x \in \mathbb{Z}_p$ and $\{x_n\}$ such that, $x_n \equiv x_{n+1} (mod \, p^{n+1})$; $0 \leq x_n \leq p^{n+1}-1$ ; $|x-x_n|_p \rightarrow 0, n \rightarrow \infty$.
...
5
votes
0answers
46 views
Why are the p-adic integers a linearly ordered group? [duplicate]
In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group.
I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
4
votes
3answers
302 views
How many $p$-adic numbers are there?
Let $\mathbb Q_p$ be $p$-adic numbers field. I know that the cardinal of $\mathbb Z_p$ (interger $p$-adic numbers) is continuum, and every $p$-adic number $x$ can be in form $x=p^nx^\prime$, where ...
4
votes
4answers
219 views
Divergent series and $p$-adics
If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct.
Surely ...
4
votes
4answers
100 views
Solution of $x^2 = 2$ in $\mathbb{Q}_p$
Let $p \neq 2$. Show that the equation
$$x^2=2 \quad (*)$$
has a solution $x \in \mathbb{Q}_p$ iff exists $y \in \mathbb{Z}$ such that $y^2 \equiv 2 \, \pmod p$.
$\Rightarrow$ Let $x= y + x_1 p + x_2 ...
4
votes
2answers
403 views
Roots of unity in a local field
The multiplicative group of a local field $K$ with valuation ring $\mathcal{O}$ and residue class field of $\overline{K}$ of degree $q=p^f$ splits as
$K=\langle \pi\rangle\times \mu_{q-1}\times ...
4
votes
3answers
79 views
How to show that $\mathbb{Q}_p$ cannot be ordered?
I've seen many references on this site and others to the fact that the $p$-adic numbers cannot be ordered, but the closest I've seen to a proof of this is Wikipedia's vague reference to ...
4
votes
2answers
178 views
Can I build a finitely additive function on $\mathbb{Q}_p$?
This is partially motivated by a question I saw earlier here, Does such a finitely additive function exist?
I've been reading about the topology of $\mathbb{Q}_p$ in Knapp's Advanced Algebra in ...
4
votes
1answer
245 views
Is there a $p$-adic version of the Riemann hypothesis?
Is there a $p$-adic version of the Riemann hypothesis or this does not make any sense?
4
votes
2answers
128 views
Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?
For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...
4
votes
1answer
139 views
The polynomial $x^p - x -1/p$ over $\mathbb{Q}_{p}$
I know that the polynomial $f(x) = x^p -x - \frac{1}{p} \in \mathbb{Q}_{p}[x]$ is irreducible. So, let $\alpha$ be a root of $f(x)$, and $K = \mathbb{Q}_p(\alpha)$. Let $O_K$ be the valuation ring of ...
4
votes
1answer
71 views
When a number is a square in the p-adic rationals - proof question (Quadratic Residues)
I'm a little stuck with the proof of a theorem I'm trying to understand. The theorem is as follows:
"For odd prime $p$, suppose for $\alpha \in Q_{p}$ (the p-adic rationals) that $|\alpha|_p=1$. Then ...
4
votes
1answer
175 views
Multiplicative Haar measure on $\mathbb{Q}_p$?
I have read in a book that if one takes $\mu$ to be the additive Haar measure on $\mathbb{Q}_p$, the p-adic rationals, then
$$\nu(A) := \int_{A} 1/|x|_p dx$$
is a multiplicative Haar measure on ...
4
votes
1answer
147 views
Is there a $p$-adic version of Liouville theorem?
That is, if a function $f$ is analytic and bounded in all $K$, a $p$-adic field (or more generally a complete non-archimedean field), has to be constant?
And does the theorem work for functions on ...
4
votes
1answer
123 views
Decomposition of $\mathbb{C}_p^*$
I'm looking for a topological group decomposition of $\mathbb{C}_p^*$. I know that I can write
$\mathbb{C}_p^*\cong p^\mathbb{Q}\times \mathcal{O}_{\mathbb{C}_p}^* \cong p^\mathbb{Q}\times ...
4
votes
1answer
46 views
what are the p-adic division algebras?
Is there a classification of division algebras over $\mathbb{Q}_p$? There are field extensions of $\mathbb{Q}_p$, but are there any others? In particular, I want to know if they are all commutative.
...
4
votes
1answer
92 views
Choosing an isomorphism $\tau:\overline{\mathbf{Q}}_p\simeq\mathbf{C}$; how do things depend on choice of $\tau$?
I sometimes see arguments that begin by choosing an isomorphism of fields $\tau:\overline{\mathbf{Q}}_p\simeq\mathbf{C}$, and then defining some property in terms of this isomorphism. I'm not so ...
