0
votes
0answers
31 views

$p$-completion of a $\mathbb{Z}_p$-module

Let $p$ be a prime number, $\mathbb Z_p$ the ring of $p$-adic integers. Let $M$ be a finitely generated $\mathbb Z_p$-module and $\widehat{M}$ its $p$-completion $\varprojlim_n M/{p^n M}$. ...
3
votes
1answer
26 views

Examples where there is no power integral basis

Let $A$ be a completion discrete valuation ring with quotient field $K$, $L/K$ finite and separable, and $B$ the integral closure of $A$ in $L$. Let $P, \mathfrak P$ be the unique maximal ideals of ...
0
votes
0answers
32 views

$P-adic$ number theory problem

Let $F(x,y,z)=5x^2+3y^2+6z^2+8xy+6xz$. Find all the rational integers $x,y,z$ such that they are not all divisible by $7$ and that $F(x,y,z)=0 mod(7^2)$. Hint: Use Hensel's lemma. Need help. ...
0
votes
0answers
23 views

Proving the set $\mathbb{Z}_2$ of 2-adic integers is compact. [duplicate]

I am currently working on the problem: Prove the set $\mathbb{Z}_2$ of 2-adic integers is compact My idea is to prove this via sequential compactness. So far, I have considered a sequence ...
0
votes
1answer
56 views

7-adic series expansion of square root of 2

Given the sequence $\{ a_n\}$ defined by the (positive and $a_n < 7^n$) solutions of the congruence $x^2 \equiv 2 \mod 7^n$ and $a_{n+1}\equiv a_n \mod 7^n$. e.g. the first one is $a_1 =3$ the ...
1
vote
1answer
53 views

Is $f(x)=x^{4}-2x^2 +3$ Eiseinstein in 2-adic $\mathbb{Q}_{2}$?

I think it is because $|1|_{2}=1$, $|2|_{2}=2^{-1}\leq 1$ and $|3|$,where 3 is prime I am using the following Eisenstein criterion for $f(x)=a_{n}x^{n}+...+a_{0}$: $|a_{n}|=1$, $|a_{0}|=prime$ and ...
0
votes
1answer
23 views

Possible Quadratic extensions of $\mathbb{Q}_{2}$ are 1-1 with $\mathbb{Q}_{2}^{*}/(\mathbb{Q}_{2}^{*})^{2}$

Why do they have to be units? $\mathbb{Q}_{2}[\sqrt{2}]$ is a quadratic extension but $|2|_{2}=\frac{1}{2}\neq 1$. Where do I need them to be units below? ...
5
votes
2answers
85 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 ...
3
votes
0answers
28 views

GL$_2(\mathbb{Q}) Z_{\mathbb{R}}$ closed in GL$_2(\mathbb{A})$?

I am struggling with the following subgroups of GL$_2(\mathbb{A})$ where $\mathbb{A}$ is (the topological ring of) Adeles over $\mathbb{Q}$: $$G_\mathbb{Q} := \iota(\text{GL}_2(\mathbb{Q})) $$ where ...
2
votes
1answer
50 views

Rational Number Form

I was reading Rational Points on Elliptic Curves by Silverman and Tate and they state: Every non-zero rational number may be uniquely written in the form $\frac{m}{n}p^v$, where $m,n$ are integers ...
3
votes
0answers
80 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 ...
2
votes
2answers
106 views

$p$-adic valuation.

Let $\alpha_1,\alpha_2\in \mathbb Z_p$ such that $v_p(\alpha_1)<v_p(\alpha_2).$ How to prouve that $v_p(\alpha_2-\alpha_1)=v_p(\alpha_1)$ ? I think this is a stupid question but I'm really ...
1
vote
0answers
41 views

What do they mean by this: “the definition of product topo shows that this mapping is continuous”

Let $\Bbb{Z}_2$ be the $2$-adic integers. There's a bijection $\psi : \Bbb{Z}_2 \to C$, the Cantor set, defined by $\psi (\sum_{i \geq 0} a_i p^i) = \sum_{i \geq 0} \dfrac{2a_i}{3^{i+1}}$. The text ...
7
votes
4answers
127 views

How do you prove that $\Bbb{Z}_p$ is an integral domain?

Let $\Bbb{Z}_p$ be the $p$-adic integers given by formal series $\sum_{i\geq 0} a_i p^i$. I'm having trouble proving that it's an integral domain.
2
votes
1answer
72 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 ...
3
votes
1answer
106 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
62 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
203 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
117 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 ...
3
votes
1answer
95 views

Why $\operatorname{Gal}(\mathbb{Q}_q(\zeta_p)/\mathbb{Q}_q)$ is isomorphic to $\mathbb{Z}/f\mathbb{Z}$

I would like to ask a question on this answer. I don't understand why $\operatorname{Gal}(\mathbb{Q}_q(\zeta_p)/\mathbb{Q}_q)$ is isomorphic to $\mathbb{Z}/f\mathbb{Z}$, where $f$ is the degree of the ...
1
vote
1answer
101 views

A non-Archimedean norm definition can be strengthened.

See Andrew Baker's p-adic notes: For a non-Archimedean norm $N$ it is true that "$N(x + y) \leq \max\{N(x), N(y)\}$, with equality if $N(x) \neq N(y)$." Having trouble proving this. Please ...
3
votes
1answer
86 views

How can every $p$-adic integer be the limit of a sequence of non-negative integers?

See Andrew Baker's P-adic Notes. Every element of $\mathbb{Z}_p = \{a \in \mathbb{Q}_p : |a|_p \leq 1 \}$ is a limit of a sequence of non-negative integers, with respect to the $|\cdot|_p$ norm. How ...
-1
votes
1answer
55 views

Finitely generated submodule of $p$-adic module is direct summand? [closed]

Are finitely generated submodules of a $p$-adic module direct summands?
1
vote
1answer
168 views

A characterization of the module function on a locally compact division ring

References: Weil's Basic Number Theory(written as BNT). Bourbaki's Commutative Algebra(written as BCA). Let $K$ be a topological ring with an identity. Suppose every non-zero element of $K$ is ...
14
votes
3answers
453 views

An automorphism of the field of $p$-adic numbers

Is an automorphism of the field $\mathbb{Q}_p$ of $p$-adic numbers the identity map? If yes, how can we prove it? Note:We don't assume an automorphism of $\mathbb{Q}_p$ is continuous.
4
votes
2answers
92 views

Proving if $|a|_p=1$ then $a$ is invertible in $\mathbb{Z}_p$

I decided to take a look $p$-adic integers. I am trying to show that $$a \in \mathbb{Z}_p \text{ is invertible if and only if } |a|_p=1$$ where $$|x|_p= \left\{ \begin{array}{ll} p^{-n} & ...
5
votes
2answers
166 views

$p$ prime, $P = \left\{ \frac{m}{p^e} \middle| m, e\in \mathbb{Z} \right\}$. Prove that $\mbox{Ext}(P; \mathbb{Z}) \cong \mathbb{Z}^{(p)}/\mathbb{Z}$

I don't know why the book Homology by Saunders Mac Lane is wwaaayyy tttoooo hard to digest. :((( This is like the third time I read this book, but still not clear is everything, and to tell the ...
1
vote
1answer
181 views

An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let K be a field, and let $|\cdot|_{1},\cdots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
7
votes
0answers
78 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
3
votes
2answers
124 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?
5
votes
0answers
54 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 ...
0
votes
0answers
89 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
216 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 ...
4
votes
2answers
90 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 ...
8
votes
2answers
354 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 ...
12
votes
2answers
325 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 ...
8
votes
1answer
112 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, ...
4
votes
4answers
166 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 ...
8
votes
2answers
232 views

p-adic numbers and $\mathbb{F}_p$

As you probably know, there is a morphism of rings : $\mathbb{Z}_p\longrightarrow \mathbb{Z}/p\mathbb{Z}$ which sends a formal sum $\sum_{i\geq 0}a_ip^i$ to $a_0$ (here $\mathbb{Z}_p$ is the ring of ...
2
votes
1answer
201 views

Roots of unity in $\mathbb Q_p$

Is there a way to get the number of solutions of an equation like $x^n=1$ in the p-adic field $\mathbb Q_p$, where $p$ is a prime and $n$ a positive integer? I know that for $n=p-1$ there are $p-1$ ...
5
votes
1answer
115 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 ...
3
votes
2answers
233 views

Roots of 1 in $\mathbb Q_p$

How to prove, that all roots of 1 in $\mathbb Q_p$ are roots of $x^{p-1}-1$? If we consider the ring homomorphism $$ \mathbb Z_p \to \mathbb F_p^*, $$ then we see, that all the roots in power $p-1$ ...
11
votes
2answers
300 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 ...
5
votes
1answer
251 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 ...
3
votes
1answer
405 views

Newton polygons

This question is primarily to clear up some confusion I have about Newton polygons. Consider the polynomial $x^4 + 5x^2 +25 \in \mathbb{Q}_{5}[x]$. I have to decide if this polynomial is irreducible ...
2
votes
2answers
240 views

Representations of p-adic integers as certain infinite sums

One way to define the p-adic integers is as the $p$-adic completion of $\mathbb{Z}$. With some additional work, it can be shown that this is isomorphic to $\mathbb{Z}[[x]]/(x-p)$. Now, I know that ...
0
votes
1answer
110 views

Characterizing $\mathbb{Q}_p$ as an $I$-adic completion of $\mathbb{Q}$

It's known that the ring of p-adic integers $\mathbb{Z}_p$ can be characterized as the I-adic completion of $\mathbb{Z}$ for $I=(p)$. Is there any similar characterization for $\mathbb{Q}_p$ (i.e. an ...
9
votes
2answers
649 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 ...
3
votes
1answer
231 views

Weird quotient of $\langle\mathbb Q,+\rangle$?

after looking at this question I came to think on one particular case. I'm wondering if maybe I've missed something on the way. If anyone could give it a look that would be great: We start by ...
7
votes
2answers
323 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 ...