The p-adic-number-theory tag has no wiki summary.
5
votes
3answers
29 views
Tensor product of a number field $K$ and the $p$-adic integers
In a paper by J. Jones and D. Roberts (http://math.la.asu.edu/~jj/localfields/database.pdf) we are introduced to an isomorphism $K \otimes \mathbb{Q}_p \cong \prod\limits_{i=1}^g K_{p,i}$, where $K$ ...
5
votes
1answer
38 views
Image of the Norm on a Finite Dimensional Extension of $\mathbb{Q}_p$
I've been trying to see whether following assertion is true in order to give a quick proof of another problem I was doing: if $K$ is a finite dimensional extension of the $p$-adic numbers ...
3
votes
1answer
29 views
When is a $p$-adic unit an $m$-th power
Given $x\in\mathbb{Q}_p^*$, we can write $x=p^nu$ where $n\in\mathbb{Z}$ and $u\in\mathbb{Z}_p^*$. Then we can decide whether or not $x$ is a square by looking at $n$ and $u$. If $p\neq 2$ then $x$ is ...
5
votes
1answer
40 views
Extension of valuation to the algebraic extension of a number field.
I am trying to get the idea how we can extend the $p$-adic valuation on $\mathbb Q$ to an algebraic extension. In particular, how to extend the $p$-adic valuation for $p = 5$ from $\mathbb Q$ to ...
5
votes
0answers
49 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 ...
2
votes
0answers
22 views
Maximal compact subgroup of $GL_n(\mathbb C_p)$
It is known that the general linear group $GL_n(\mathbb Q_p)$ over the $p$-adic numbers has $GL_n(\mathbb Z_p)$ as a maximal compact subgroup and every other maximal compact subgroup of $GL_n(\mathbb ...
4
votes
0answers
31 views
Gelfand's formula, different field
Gelfand's formula says that for a complex matrix $A \in \mathbb{C}^{n \times n}$, $$\rho(A) = \lim_{m \rightarrow \infty} \|A^m\|^{1/m},$$ where $\rho$ is the spectral radius (norm of maximal ...
4
votes
0answers
33 views
Group structure of $\mathbb{Q}_p ^* / \mathbb{Q}_p ^{*3}$
Let p be 1 mod 3 (separate question: work out 2 mod 3). What is the group structure of the abelian group $\mathbb{Q}_p ^* / \mathbb{Q}_p ^{*3}$?
$\mathbb{Q}_p ^*$ refers to the group of units in ...
1
vote
1answer
36 views
Prove minimal polynomial over $\mathbb{Q}$ is reducible over $\mathbb{Q}_p$ and $\mathbb{R}$
Let $f$ be the minimal polynomial of $\alpha = \sqrt{-1} + \sqrt{17} + \sqrt{-17}$ over $\mathbb{Q}$ (with degree 4). Prove $f$ is reducible over $\mathbb{Q}_p$ (p-adic rationals) for all primes p and ...
0
votes
0answers
73 views
all various cubic extensions of Q7
I need to classify all various cubic extensions of $\mathbb Q_7$? How can one do it?
2
votes
1answer
41 views
p-adic isomorphism $\mathbb{Q}_p\not\cong \mathbb{Q}_q\iff p\ne q$ [duplicate]
In class I learn that $\mathbb{Q}_3\not\cong\mathbb{Q}_5$ because one of them has $\sqrt{2}$ the other doesn't. Also professor asks us to find reference that $\mathbb{Q}_p\not\cong \mathbb{Q}_q\iff ...
4
votes
3answers
80 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 ...
2
votes
1answer
29 views
How to find $\sup(\{|x-y|_p : x,y\in B(0;r)\})$
Just to clarify the notation and the question:
Working in p-adic space $\mathbb{Q}_p$, we have the norm $|x|_p=p^{-ord_p(x)}$ and we define the metric over this space as $d(x,y)=|x-y|_p$. We are ...
1
vote
0answers
32 views
Irreducible Polynomials over $\mathbb{Q}_p$
I was reading a paper and the author had an irreducible quadratic polynomial $f(x)$, with (non-real) root $\alpha$. He stated that if $p$ ramified or stayed prime in $\mathbb{Q}(\alpha)$ then $f(x)$ ...
1
vote
0answers
47 views
Representing a fraction as a $p$-adic number
If we have the following $p$-adic number:
$$2+3p+5p^2+2p^3+3p^4+5p^5+2p^6+3p^7+5p^8+.....$$ and I am trying to find what rational number this p-adic number represents. I have no idea as to how to go ...
5
votes
5answers
177 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 ...
3
votes
2answers
53 views
Finding root using Hensel's Lemma
Hensel's Lemma calculates root of a polynomial $\in \mathbb{Z}_p[X]$ but is there any other significance to other branches of mathematics or outside mathematics? Why is finding root of ...
0
votes
0answers
24 views
Nonnegative sequence which relates to $p-$adic analysis
Well I face to the following problem: Let $p$ is a fixed prime number ($p=2,3,5,7,\ldots$).
I am trying to find non-negative sequence $(a_n)_{n\geq0}$ satisfies two condition
(a) ...
2
votes
0answers
45 views
$p$-adic Integration
I am struggling to compute the following integral
\begin{equation}
\int_{\mathbb{Z}_p}\exp_p(|x|_p)d\mu
\end{equation}
where
\begin{equation}
\exp_p(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!} \text{ defined ...
1
vote
2answers
55 views
Irrational P-adics
$\mathbb{Q}_p$ is completion of $\mathbb{Q}$ by defining a new metric. So, with respect to this new metric they are complete.
I just want to be sure, are there p-adic rationals? If there are P-adic ...
2
votes
1answer
38 views
congruence modulo infinity
Going through Hensel's Lemma, I feel I read somewhere that the limit of sequence of integers $a_0,a_1,a_2,...$=$ a$ is root of the $f(X)\in\mathbb{Z}_p[X]$, where,
...
1
vote
1answer
69 views
Functional equation on $\mathbb Q_p^\star$
I am trying to find solutions of functional equation $f(xy)=f(x)+f(y)$ for all $x,y\in\mathbb Q_p\setminus\{0\}=\mathbb Q_p^\star$. Where $f:\mathbb Q_p^\star\to\mathbb R$. I know some solutions:
1) ...
4
votes
3answers
144 views
Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$
Let G be the multiplicative group $\mathbb{Z}_n^*$ for $n = 2^k$ and $k \ge 3$.
Can we prove that no element has order bigger than $2^{k-2}$ ?
My solution (not really a solution) :
Since $n=2^k$, I ...
2
votes
1answer
34 views
What is a group of type $(2,2)$?
This is a statement in Serre's A Course in Arithmetic (p. 18).
If $p\ne2$, the group $\mathbb{Q}_p^*/\mathbb{Q}_p^{*2}$ is a group of
type $(2,2)$.
What is a group of type $(2,2)$?
-4
votes
1answer
120 views
Verifying that $\frac{p^n}{p^{n}-1}$ converges $p$-adically to $0$, while $\frac{1}{p^{n}-1}$ converges $p$-adically to $1$
This is a question from a book I'm struggling with, please could you provide a clear proof?
Fix a prime number $p$. Verify that $\dfrac{p^n}{p^{n}-1}$ converges $p$-adically to $0$, while ...
1
vote
0answers
68 views
By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?
this is a question from a book I'm struggling with, please could you provide a clear proof
Consider the sequence of rational numbers
$a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1
...
2
votes
1answer
100 views
For what primes $p$ does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?
this is a question from a book I'm struggling with, please could you provide a clear proof
For what primes p does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?
kind thanks
0
votes
1answer
55 views
For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically
For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically and, when it does, to what limit?
0
votes
0answers
96 views
Show that the field of p-adic numbers is complete
this is a question from a book I'm struggling with, please could you provide a clear proof
Show that the field of p-adic numbers is complete
i.e. that a sequence of p-adic numbers converges if and ...
0
votes
1answer
26 views
p- adic isomorphims
Let $m_p=\{ x \in \mathbb{Q} : v_p(x)>0\}$, $v_p$ $p$-adic valuation. Let $k >0 $ a integer and the ideal $I=m_p^k$. Then,
$$O_p/ I \cong \mathbb{Z}/p^k \mathbb{Z} $$
with, $O_p=\{ x \in ...
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 ...
5
votes
1answer
58 views
p-adic expansion
Let $x \in \mathbb{Z}_p$ and $\{x_n\}$ such that, $x_n \equiv x_{n+1} (mod \, p^{n+1})$; $0 \leq x_n \leq p^{n+1}-1$ ; $|x-x_n|_p \rightarrow 0, n \rightarrow \infty$.
...
2
votes
3answers
25 views
p-adic norm, Sets equal
Let $p$ a prime, and the p-adic norm $|x|_p = (\frac{1}{p})^{v_p(x)}$, with $v_p$ the p-adic valuation. Show that
$$\{ |x|_p : x \in \mathbb{Q}_p \} = \{ p^k : k \in \mathbb{Z}\}$$
My question is ...
2
votes
2answers
36 views
Valuations, Isomorphism, Local ring
Let $x \in \mathbb{Q}, \, x \neq 0$, such that $x=p^r \frac{a}{b}, \quad a,b, r \in \mathbb{Z}, \quad p \nmid a, \quad p\nmid b$.
Let $v_p(x):=r$ and $v_p(0):= \infty$. Also,
$$\mathcal O_p= \left\{ ...
1
vote
2answers
84 views
P-adic numbers complete/incomplete
P-adic numbers are complete in one sense and incomplete in another sense. Is it so?
Firstly, does not complete mean connected? I read somewhere that there is not intermediate value theorem for ...
4
votes
2answers
131 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 ...
3
votes
0answers
52 views
Why is the Euclidean metric called the prime at infinity?
I've been studying p-adic analysis recently and after a bit of searching on the web, I haven't found an answer as to why the Euclidean metric is referred to as the 'prime at infinity', and given the ...
5
votes
1answer
86 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 ...
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, ...
2
votes
2answers
80 views
Questions regarding p-adic expansion and numbers
As opposed to real number expansions which extend to the right as sums
of ever smaller, increasingly negative powers of the base $p$,
$p$-adic numbers may expand to the left forever, a property ...
3
votes
0answers
35 views
Spectrum theorem for p-adic matrix analysis
Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way: 1: whether there exist spectrum theorem for p-adic matrix.\
2: ...
4
votes
1answer
47 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.
...
2
votes
2answers
60 views
Prove that the sequence satisfying $a^{p^n}\equiv a^{p^{n-1}}$mod $p^n$ for $n\geq 1$, converges to the Teichmuller Representative
Prove that for any $a\in\mathbb{Z}_p$, the sequence satisfying $a^{p^n}\equiv a^{p^{n-1}}$mod $p^n$ for $n\geq 1$, converges to the Teichmüller Representative congruent to $a$ mod $p$.
So a p-adic ...
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 ...
1
vote
0answers
68 views
Necessary and sufficient conditions for Hensel lifting in the multidimensional case
in Multidimensional Hensel lifting, @Hurkyl gave a neat sufficient condition for the existence of $p$-adic liftings in the multidimensional case. I have finally gotten around (but please also see ...
1
vote
2answers
77 views
Prove that $\displaystyle\mathbb{Q}_p$ always contains $p$ solutions to the equation $x^p-x=0$
Prove that $\mathbb{Q}_p$ always contains $p$ solutions $a_0,a_1,...,a_{p-1}$ to the equation $x^p-x=0$ satisfying $a_j\equiv j$ (mod p).
This is an intense problem as far as I'm concerned. I'm not ...
2
votes
2answers
94 views
Is the set of integers with respect to the p-adic metric compact?
Given the integers and a prime $p$. I thought I had successfully shown that $\mathbb{Z}$ was compact with respect to the metric $|\cdot |_p$, by showing that the open ball centered at zero contained ...
3
votes
0answers
50 views
p-adic liftings on SAGE
I asked a question the other day:
Multidimensional Hensel lifting
which @Hurkyl kindly and very elegantly answered. A follow-on from this is that I have tried to implement exactly the "algorithm" ...
3
votes
4answers
73 views
Profinite and p-adic interpolation of Fibonacci numbers
On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here)
For each profinite integer $s$, one can in a natural way define the $s$th Fibonacci ...
3
votes
1answer
145 views
Multidimensional Hensel lifting
I have a question about a practical application of (some) generalised form of Hensel's Lemma. I cannot find it stated in an appropriate form in Bourbaki or anywhere else, so here goes ...
Let $p$ be ...
