1
vote
2answers
34 views

A quadratic equation over $\mathbb{Q}_p$

Suppose we have the equation $x^2+x+1$ over the field $\mathbb{Q}_p$. is it possible to determine for what primes $p$ the equation has solutions? I tried to see whether this is related to what $p$ is ...
5
votes
1answer
59 views

Problems that are made easy by using p-adic numbers

Does anybody know of elementary problems that can be be solved using the p-adics? Solutions are preferred.
1
vote
0answers
29 views

Open Ball under the p-adic Norm

I'm trying to figure how, if it's even possible, to draw an open ball using the p-adic norm. My definition of the p-adic norm I'm using is: $ \lvert x \rvert_p $ = $p^{-ord_px}$ if $x \neq 0$ and ...
1
vote
1answer
15 views

Ramification and roots of unity in complete discrete valuation rings.

Let $\mathcal{O}$ be a complete discrete valuation ring with algebraically closed residue field $k$ of characteristic $p>0$. Let $\pi\in \mathcal{O}$ generate the maximal ideal and suppose ...
4
votes
2answers
59 views

Degree of closure of $\mathbb{Q}_p$

In order to prove that algebraic closure of $\mathbb{Q}_p$ is infinite, I took the polynomial $x^n-p$ with $n>1$ over $\mathbb{Q}_p$ to show that this eqaution has no solution for infinite cases to ...
0
votes
1answer
43 views

Embedding into $p$-adic complex numbers

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind: Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the ...
5
votes
4answers
80 views

Can one show a beginning student how to use the $p$-adics to solve a problem?

I recently had a discussion about how to teach $p$-adic numbers to high school students. One person mentioned that they found it difficult to get used to $p$-adics because no one told them why the ...
0
votes
1answer
41 views

Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In ...
3
votes
2answers
70 views

Let $p \equiv 2 \mod{3}$. For any $a \in Z$ such that $ p \nmid a$ , show that there exists $x \in \mathbb{Z}_p$ with $x^3 = a$.

Let $p \equiv 2 \mod{3}$. For any $a \in Z$ such that $ p \nmid a$ , show that there exists $x \in \mathbb{Z}_p$ with $x^3 = a$. I've tried using Hensel's lemma and the fact that if $p \equiv 2 ...
3
votes
0answers
76 views

An application of Hensel's lemma to some different fields

I would like to prove the following. Let $p>2$ be a prime number, $\mathbb{Q}_{p}$ the field of p-adic numbers Let $u\in \mathbb{Z}_{p}^{\times}$ be a unit. (1)Prove that the following are ...
1
vote
1answer
44 views

Embedding $\mathbb Q^c$ into $\mathbb Q^c_p$

Let $p$ be a prime number and $\mathbb Q_p$ the $p$-adic completion of $\mathbb Q$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. Is there an embedding $$j: \mathbb Q^c ...
2
votes
1answer
43 views

Definition of $\mathbb Q^c_p$

Let $p$ be a prime number and let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q $ in $\mathbb C$, i.e. the field of algebraic numbers. Is it possible at all to define the $p$-adic completion ...
2
votes
1answer
68 views

$Q_p(\zeta)$ where $\zeta$ is a $p$-th root of $1$.

I'm not looking for a full solution, only a hint please! Let $\zeta$ be a $p$-th root of unity in an algebraic closure of $Q_p$. Show that $Q_p(\zeta) = Q_p ((-p)^{\frac{1}{p-1}})$. Following a hint ...
1
vote
0answers
52 views

Arithmetic in p-adic

I came to the problem saying that there exists a number $x\in \Bbb Z_7$ such that $x^2=2$ but there is no such $x$ in $\Bbb Z_5$. Could anyone give an explanation of this? How to actually find the ...
7
votes
1answer
92 views

Local solutions over $\mathbb{Q}_p$ but no solutions over $\mathbb{Q}$

I was looking at a set of notes that states the equation $x^4-17=2y^2$ is solvable locally over $\mathbb{Q_p}$ for every $p$ , but is not solvable over $\mathbb{Q}$. Now, this is not a homework ...
1
vote
1answer
97 views

non integer p adic expansion (special case)

I need to calculate the 5 adic expansion of $\frac{1}{45}$. Since i cannot compute it normally, i expand $\frac{1}{45}$ into $\frac{1}{5}*\frac{1}{9}$. I calculated the 5 adic expansion of ...
2
votes
3answers
174 views

P-adic expansion construction

Can anyone teach me about p-adic expansion? especially the case where we have to expand a square root. I need to know how to construct them. for example: the 7-adic expansion of $\sqrt{305}$. This ...
1
vote
1answer
326 views

P-adic integers and roots of unity

Show that $\Bbb Z_p$ contains all the $(p-1)$th roots of unity. For which primes $p$ does $\Bbb Z_p$ contains primitive fourth roots of unity? Here $\Bbb Z_p$ is the ring of $p$-adic integers. Proving ...
0
votes
0answers
43 views

Ring of integers of unramified extension

Let $L/K$ be unramified extension of local fields, and $k,l$ - their residue fields, $l=k(\overline \alpha)$. Is it true that $\mathcal O_L=\mathcal O_K[\alpha]$? And can it be proved if it's true.
3
votes
0answers
48 views

$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$?

Let $K$ be a totally ramified extension of $\mathbb Q_p$ of degree $n$. Then $$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n) .$$ What is this isomorphism?
1
vote
0answers
74 views

Hensel's lemma in $\mathbb Z_2$

Can you give me a concrete example for a quadratic form $$ f(x,y)=ax^2+bxy+cy^2 \in \mathbb Z_2[x,y] $$ which has a primitive solution $(x^*,y^*) \in \mathbb Z_2 \times \mathbb Z_2$ (mod 4) with the ...
3
votes
1answer
81 views

Properties of squares in $\mathbb Q_p$

Let $\mathbb Q_p$ be the field of $p$-adic numbers. I know that for $p \neq 2$ an element $x=p^n u \in \mathbb Q_p^\times$ (with $n \in \mathbb Z$ and $u \in \mathbb Z_p^\times$) is a square if and ...
2
votes
1answer
68 views

Discrete valuation on $p$-adic numbers

For the ring of $p$-adic integers $\mathbb Z_p$ let $\mu_n: \mathbb Z_p \to \mathbb Z / p^n \mathbb Z$ be the projection mapping. Consider $\mathbb Z / p^n \mathbb Z$ with the discrete topology. Is ...
2
votes
1answer
56 views

Structure of $(\mathbb{Z}/p^k\mathbb{Z})^\times$ other than via $p$-adics?

I think I heard somewhere long ago that $\mathbb{Z}/p^k\mathbb{Z}$'s unit group is cyclic if $p$ is odd and $C_2\times C_{2^{k-2}}$ if $p=2$. I remember trying to prove it and finding it surprisingly ...
3
votes
2answers
178 views

Ring of $p$-adic integers $\mathbb Z_p$

There are a few ways to define the $p$-adic numbers. If one defines the ring of $p$-adic integers $\mathbb Z_p$ as the inverse limit of the sequence $(A_n, \phi_n)$ with $A_n:=\mathbb Z/p^n \mathbb ...
7
votes
1answer
116 views

A valuation-like function $w: \mathbb{N}^{+} \rightarrow \mathbb{N}$ is a $p$-adic valuation?

This question is a variant of problem 4, pg. 21, from Birkhoff and Maclane, A Survey of Modern Algebra. Given a function $w: \mathbb{N}^+ \rightarrow \mathbb{N}$ that behaves like a valuation ...
18
votes
1answer
232 views

Why is there “no analogue of $2i\pi$ in $\mathbf C_p$”?

In his paper Fonctions L p-adiques, Pierre Colmez says: Tate a montré qu'il n'existait pas dans $\mathbf C_p$ d'analogue $p$-adique de $2i \pi$ et donc par conséquent que les périodes $p$-adiques ...
3
votes
1answer
97 views

Generators of Zp* and p-adic cyclotomic character

Let $p$ be an odd prime number. It is known that $\mathbb{Z}_p^{\times}$ is topologically cyclic. Now let $\chi_{\mathrm{cyclo}} : Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p) \to \mathbb{Z}_p^{\times}$ be ...
8
votes
1answer
218 views

Galois group over $p$-adic numbers

Can one describe explicitly the Galois group $G=\operatorname{Gal}(\overline{\mathbb Q_p}/\mathbb Q_p)$? I only know the most basic stuff: unramified extensions of $\mathbb Q_p$ are equivalent to ...
3
votes
0answers
36 views

Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
1
vote
1answer
88 views

On the numerators of Bernoulli numbers

Von Staudt-Clausen theorem implies that $pB_{2n} \in \mathbb{Z}_{p}$ for all primes $p$ and for all $n \in \mathbb{N}$. It means that the highest power of any prime that can occur in the denominator ...
3
votes
1answer
209 views

Characterization of integers which has a $2$-adic square root

Does anyone know an "elementary" proof of the following theorem? Let $k \neq 0$ be a rational integer. Then $k$ admits a square root in $\mathbb{Z}_2$ if $k = 4^a (8b+1)$ for some $a \in \mathbb{N}$, ...
2
votes
3answers
427 views

p-adic expansion of a rational number

Studying $p$-adic numbers I encountered the following theorem: Given a eventually periodic sequence $(a_n)_{n=k}^{\infty}$ such that $0 \le a_n <p$, the sum \begin{equation*} ...
6
votes
1answer
96 views

Relations between irreducibility on $\mathbb{Q}[x]$, and on $\mathbb{Q}_p[x]$ ($p$-adic numbers)

I'm reading "$p-$adic numbers: An introduction" by Fernando Q.Gouvêa, and I'm currently on page 79 of the book. Problem 121. Show that the equation $(X^2 - 2)(X^2 - 17)(X^2 - 34) = 0$ has a root ...
3
votes
3answers
97 views

Raising a rational integer to a $p$-adic power

Let $p$ be a prime number and $d$ a positive integer. If $n\in \mathbb{Z}_p$ (the ring of $p$-adic integers) how would one define $d^n$? Under what conditions would this be an element of ...
1
vote
1answer
63 views

p-adic Eisenstein series

I'm trying to understand the basic properties of the p-adic Eisenstein series. Let $p$ be a prime number. Define the group $X = \begin{cases} \mathbb{Z}_p\times \mathbb{Z}/(p-1)\mathbb{Z} & ...
1
vote
0answers
51 views

Ring of the residual classes $(\Bbb Z/p\Bbb Z)^\times$? $p$-adic integer?

In a recent question we raised the theorem: for a given prime $p$ and a given power $m$ the representation of any positive integer $n\in \Bbb N$ in the form: $$ n=(a_u p - b_u) \; p^m$$ is unique ...
1
vote
1answer
73 views

connect p-adic expansion and fundamental theorem of arithmetic?

On the way to explain a $p$-adic expansion, we consider, when dealing with natural numbers, if we take $p$ to be a fixed prime number, then any positive integer expansion in the form can be written as ...
0
votes
0answers
39 views

Is there an intuition behind $ n=(a_u p - b_u) \; p^m$ from p-adic number theory pov?

In a recent question I conjectured that: for a given prime $p$ and a given power $m$ the representation of any positive integer $n\in \Bbb N$ in the form: $$ n=(a_u p - b_u) \; p^m$$ is unique with ...
8
votes
1answer
326 views

Structure of p-adic units

I am trying to understand the structure of the $p$-adic units. I know that we can write $$\mathbb{Z}_p^\times \cong \mu_{p-1} \times 1 + p\mathbb{Z}_p,$$ where $\mu_n$ are the $n$th roots of unity in ...
3
votes
2answers
111 views

Where do $p$-adic numbers and $p$-Sylow theory both appear?

Both $p$-adic numbers and $p$-Sylow theory are by design "arithmetic" ways of "localizing," so it stands to reason they might be in cahoots in certain contexts. Are they?
6
votes
3answers
149 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
51 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 ...
0
votes
0answers
88 views

all various cubic extensions of Q7

I need to classify all various cubic extensions of $\mathbb Q_7$? How can one do it?
8
votes
1answer
205 views

$S$-Units notation and Dirichlet's unit theorem

I'm having a hard time understanding some notions of a paper I'm working on. Let $L/K$ be a finite normal extension of number fields and $S$ be a set of places of $K$ prime to $p$ where $p$ denotes an ...
1
vote
0answers
69 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
271 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 ...
-4
votes
1answer
143 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 ...
0
votes
0answers
82 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 ...
1
vote
1answer
185 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