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.

learn more… | top users | synonyms

2
votes
2answers
83 views

2-adic expansion of (2/3)

I have been asked in an assignment to compute the 2-adic expansion of (2/3). It just doesn't seem to work for me though. In our definition of a p-adic expansion we have $x= \sum_{n=0}^{\infty}a_np^n$ ...
1
vote
1answer
48 views

p-adic expansions and reciprocal

Trying to get my head around p-adics,as i learn some more advanced number theory techniques but im stuck on an exercise. If we let $\alpha = a_0 + a_1p + a_2p^2 + \dots = \sum_{n=0}^\infty a_np^n$ be ...
4
votes
1answer
56 views

A question about the additive group of the $p$-adic integers

Let $J_p$ be the additive group of the $p$-adic integers. I know that it is torsion-free. I'm not pretty confortable with $p$-adic. Is it possible to find a direct sum of infinitely many cyclic ...
3
votes
1answer
117 views

The maximal unramified extension of a local field may not be complete

While reading my notes of a course in local class field theory, I arrived to a remark where it is said that given a complete discrete valuation field $K$, its maximal unramified extension $$K^{ur}= ...
1
vote
1answer
53 views

Inverse limit and union of $\mathbb{Z} / p^{n}\mathbb{Z} $

Let $p$ be a prime number and let the natural embeddings $\mathbb{Z}/p\mathbb{Z} \subset \mathbb{Z}/p^{2}\mathbb{Z} \subset \dots \subset \mathbb{Z}/p^n\mathbb{Z} \subset \dots $ Questions: Does ...
0
votes
2answers
72 views

Does $\sum n $ converge p-adically?

Does $\sum n $ converge p-adically, I have worked out $v_p(n) \leqslant log(n)/log(p) $ not sure how to conclude from this I want to prove this using the result that it converges p-adically iff ...
0
votes
1answer
59 views

Calculating p-adic valuation $v_p(n)$, using basic properties

Calculating p-adic valuation $v_p(n)$ I'm not confident with the properties of $v_p(n)$ Where $v_p(n) = $ the biggest integer $e$ such that $p^e$ divides $n$, if $n\not=0$, and $+\infty$ if $n=0$. ...
1
vote
0answers
42 views

Norm group of non-Abelian Galois extension of p-adic numbers

Let $K=\mathbb Q_p(\xi_p,\alpha)$ be an extension of $\mathbb Q_p$, where $\xi_p$ is a primitive $p$-th root of the unite and $\alpha$ is a root of $X^p=p$. Now $K/\mathbb Q_p$ is a Galois entension ...
3
votes
0answers
47 views

the different of finite extension of p-adic numbers

Let $F=\mathbb Q_p$ be the field of p-aidc numbers. Let $\xi_n$ be a $n$-th primitive root of unity. Now consider the finite extension of fields $K/F$ where $K=\mathbb Q_p(\xi_n)$. I want to find the ...
1
vote
3answers
94 views

How to find a p-adic expansion?

I've been reading about p-adic numbers recently and came across a question that asks to find the $5$-adic expansion of -3. I've been unable to find any similar examples so I can see how to work my ...
1
vote
1answer
61 views

Regarding constructing the $I$-adic completion of a ring

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For $n \geq m$, let $\varphi_{m,n}$ denote the canonical ring homomorphism $R / I^n \to R / I^m$. Let $J = \cap_n I^n$. Then $R / J$ ...
2
votes
0answers
43 views

Lattices in $\mathbb{Q}_p^n$ with the same stabilizer

Consider the action of $GL_n(\mathbb{Q}_p)$ on $\mathbb{Q}_p^n$, and let $T$ be the diagonal torus. Let $\Lambda$ and $\Lambda'$ be full-rank sublattices ($\mathbb{Z}_p$-submodules of rank $n$) such ...
1
vote
1answer
21 views

What is meant by the form of a polynomial in $A_n$ deduced from a polynomial $f$ over $\mathbb{Z}_p$?

I am reading Serre's A Course in Arithmetic and am having trouble understanding what he means by a polynomial deduced from a polynomial over $\mathbb{Z}_p$. Specifically Serre writes, ...
3
votes
0answers
26 views

Iwasawa module: $O[[G]]$ is noetherian where $G$ is a compact $p$ adic Lie group

Let $G$ be a compact $p$ adic Lie group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Then how to show that $O[[G]]$ is noetherian. I was reading the article here and on ...
1
vote
1answer
47 views

For which $a>0$ does the equation $x^2+y^2+z^2=a$ have a solution in $\mathbb{Q}_2$?

We want to check for which $a>0$ we have that the equation $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$. $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$ for $x \in \mathbb{Z}_2^{\star}, y ...
2
votes
1answer
34 views

$P$-adic Numbers Least Close to One

If I understand the definition of $p$-adic numbers, then the numbers that are $2$-adically least close to one are $3, 7, 11, \ldots$ because they are divisible by $2^1$. Do the two-adic numbers, ...
1
vote
1answer
55 views

Show that $-1=\sum_{0}^{\infty} (p-1)p^i$ in $\mathbf{Q}_p$

To show that in the field $\mathbb{Q}_p$, where $p$ is a prime, it holds that: $$-1=\sum_{0}^{\infty} (p-1)p^i$$ I did the following: It suffices to show that: $\left|\sum_0^N (p-1)p^i+1 \right|_p ...
1
vote
1answer
50 views

$f(x)=x^2-a \in \mathbb{Z}[x]$ - show propositions

Let $f(x)=x^2-a \in \mathbb{Z}[x]$. $$p \in \mathbb{P}, p \neq 2, p^2 \nmid a$$ The equation $f(x)=0$ If $p \mid a $, the equation has no solution in $\mathbb{Q}_p$ Let $p \nmid a$. The ...
1
vote
1answer
119 views

pth root of unity in $p$-adic field

It is well known that $\mathbb{Q}_p(\mu_n)$ is a totally ramified extension of degree $(p-1)p^n$ if $\mu_n$ is a primitive $p^n$th root of unity. However how true is this statement for a finite ...
1
vote
1answer
73 views

Question about the proof of Hensel's Lemma

Hensel's Lemma: Let $F(x)=\alpha_0 + \alpha_1 x+ \dots + \alpha_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p$-adic $a_1 \in \mathbb{Z}_p$ such that $$F(a_1) \equiv 0 \mod p ...
3
votes
1answer
62 views

There is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m \mid (p-1)$

After Hensel's Lemma there is the following proposition in my notes: If $p$ is a prime and $m \in \mathbb{N}$ then there is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m ...
0
votes
0answers
37 views

The definition of p-adic numbers

Are 3, 5, 7, 11, 13, 15 p-adically close to one because their differences with one are divisible by two, which is two to the first power? Possibly they are also equally distant from or close to one? ...
0
votes
0answers
15 views

The equation $P(X,Y)$ has a solution in $\mathbb{Q}_p$

Proposition Let $P(X,Y) \in \mathbb{Z}[X,Y]$. The following propositions are equivalent: The equation $P(X,Y)$ has a solution in $\mathbb{Q}_p$. For each $n \geq 0$ the equation $P(X,Y)$ has a ...
0
votes
0answers
33 views

Idea of Hensel's Lemma

$$P(x, y)=0 \tag {1} \\ \text{ where }P(x, y) \in \mathbb{Q}[x, y]$$ How do we know that $(1)$ has a solution in a $\mathbb{Q}_p$ ? We will apply Hensel's Lemma. Idea: we begin from an element ...
1
vote
1answer
30 views

$K-$rational solution of the equation - Is $\mathbb{Q} \leq \mathbb{Q}_p$?

Let $P(x, y) \in \mathbb{Q}[x, y]$. We consider the equation $P(x, y)=0$. If $a, b \in \mathbb{Q}$ such that $P(a, b)=0$ then $(a, b) \in \mathbb{Q}^2$, is called a rational solution. If $K$ a ...
1
vote
1answer
33 views

How do we deduce that $\mathbb{Q}_p=\{ p^mu\mid u \in \mathbb{Z}_p^{\star}, m \in \mathbb{Z}\} \cup \{0\}$?

We know that each element $x$ of $\mathbb{Z} \setminus{\{0\}}$ has a unique representation of the form $x=p^m u\mid m \in \mathbb{N}_0, u \in \mathbb{Z}_p^{\star}$. $$\mathbb{Q}_p=\left \{ ...
5
votes
2answers
82 views

Diophantine equation not solvable in $\mathbb{Q}$, but in $\mathcal{O}_p$

I'm trying to think of an example of a diophantine equation which can be solved in $ \mathcal{O}_p$ (meaning it can be solved $\mod p^k$ for all $ k $) for all prime $ p $'s, but not in $\mathbb{Q}$ ...
0
votes
1answer
61 views

Taylor expansion in $p$-adic integers

Let $f \in Z_p[X]$, then for $ x, y \in Z_p$, $\exists a \in Z_p$ s.t. $f(y)=f(x)+(y-x)f'(x)+(y-x)^2a$. Why is Taylor formula applicable to polynomial in $p$-adic integers $Z_p$? What condition ...
3
votes
1answer
55 views

Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
1
vote
1answer
55 views

Efficient Method/Algorithm to Compute the P-adic Ordinal of a Positive Integer

I am currently writing my master's thesis at Cal Poly Pomona, and am currently investigating the ruler sequence for a prime base. The ruler sequence for base $2$ is : ...
1
vote
1answer
53 views

Why does it suffice to show it for positive integers?

I am looking at the proof of the product formula theorem: For each $x \in \mathbb{Q}$, it holds $$\prod_{p \leq \infty} |x|_p=1$$ The proof starts by this: It is enough to show it for ...
3
votes
0answers
41 views

Herbrand Quotient: Formula for $|K^*/(K^*)^n|$

NB: Please restrict answers to hints and not solutions. Problem: Use the theory of the Herbrand quotient $q(A)=H^{0}(A)/H^{1}(A)$ to show that, if $K$ is a finite extension of $\mathbb{Q}_p$, and ...
3
votes
1answer
80 views

Explicitly computing uniformisers of local fields

Consider the field tower $L/K'/K$ where $L=\mathbb{Q_3}(\xi,2^{1/3})$, $K'=\mathbb{Q_3}(\xi) $ and $K=\mathbb{Q_3}$. Here, $\xi$ is a primitive cube root of unity, and $\mathbb{Q_3}$ is the 3-adics. ...
1
vote
2answers
55 views

Finite index subgroup $G$ of $\mathbb{Z}_p$ is open.

Suppose $[\mathbb{Z}_p:G] = n <\infty$. Write $n = p^km$ with $p\nmid m$. The idea is to show that $p^k\mathbb{Z}_p = n\mathbb{Z}_p \subseteq G$, after which I am done, since for any $x\in G$ we ...
3
votes
0answers
64 views

For $p\ge3$ there is no extension of $\mathbb Q_p$ with Galois group $S_4$

I'm trying to show that if $p\ge3$ is prime, then There is no extension $K$ of the field of $p$-adic numbers $\mathbb Q_p$ with Galois group $S_4$. I know that $K$ must have a subextension ...
3
votes
2answers
75 views

$0$ and $p^n \mathbb{Z}_p$ are the only ideals of $\mathbb{Z}_p$

I am looking at the proof of the sentence: $\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds that $\bigcap_{n \in \mathbb{N}_0 p^n \mathbb{Z}_p}=0$ ...
2
votes
1answer
141 views

Is $123456788910111121314\cdots$ a $p$-adic integer?

On the back of this question comes the natural question of whether the string $$1234567891011121314\!\cdots$$ is even a number at all. While that sort of question is vague, given the lack of generic ...
2
votes
2answers
95 views

Why is this function an embedding?

We have the canonical function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p, x \mapsto (\overline{x})_{k \in \mathbb{N}_0}=(\overline{x}, \overline{x}, \overline{x}, \dots )$. The function $\epsilon_p: ...
1
vote
1answer
96 views

Set of integer p-adics-Proposition

Proposition: "$\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds $\bigcap_{n \in \mathbb{N}_0} p^n \mathbb{Z}_p=0$ and $\mathbb{Z}_p \ ...
3
votes
1answer
36 views

$GL_2(\mathbb{Q}_p)$ and $GL_2(\mathbb{Z}_p)\cdot\mathbb{Q}_p^\times$

I am confused by a question, which is probably of school level. In some papers I have seen an induction from the group $GL_2(\mathbb{Z}_p)\cdot\mathbb{Q}_p^\times$ to the group $GL_2(\mathbb{Q}_p)$, ...
1
vote
2answers
43 views

How to prove $x^2=-1$ has a solution in $\mathbb{Q}_p$ iff $p=1\mod 4$

Let $p$ be prime and let $\mathbb{Q}_p$ denote the field of $p$-adic numbers. Is there an elementary way to prove $x^2=-1$ has a solution in $\mathbb{Q}_p$ iff $p=1\mod 4$? I need this result, but I ...
0
votes
2answers
77 views

Deriving the ultrametric from the p-adic norm?

I had thought that the ultra-metric property was just a rule that someone made up, that if applied shows some bizarre behavior. I however came across these notes: Lecture notes and it seems that the ...
1
vote
1answer
29 views

A question on hilbert symbol in $Q_p$

Let $\alpha, \beta , \gamma$ are non-zero elements of $Q_p$, show that $$(\alpha\gamma,\beta\gamma)=(\alpha,\beta)(\gamma,-\alpha\beta)$$, where $(\alpha,\beta)=1 $ or $-1$ whether $X^2-\alpha ...
0
votes
0answers
27 views

References for Hilbert symbols on $p$-adic fields

Can somebody give me some reference (Please not Serre, as it is too tough for me now) any reference for the basics and concepts on $p$-adic rings and fields and then gradually relating them to ...
0
votes
1answer
53 views

A simple question on p-adic fields

I have asked many question tonight on $p$-adic and I am still confused. So here is a very basic thing I want to know but nobody has cleared this doubt. It might be very silly, but please answer it. ...
4
votes
1answer
70 views

Which $p$-adic fields contain these numbers?

Question: Determine the $p$-adic fields which contain $$ a)\;\sqrt{-1} \qquad b)\;\sqrt{3} \qquad c)\;\sqrt{-7} \qquad d)\;\sqrt{17}$$ I have no idea on this as I am completely confused with ...
3
votes
1answer
66 views

Two doubts about squares in $\Bbb Z_p$

The statement says that for $p \neq 2$ an element $x=p^i u \in \mathbb Q_p^\times$ (with $i \in \mathbb Z$ and $u \in \mathbb Z_p^\times$) is a square if and only if $i$ is even and $u$ is a square in ...
5
votes
2answers
108 views

Why does taking completions make number fields simpler?

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example, If $K$ is a finite extension of $\mathbb ...
2
votes
1answer
56 views

Begginer doubt in Ring of p-adic integers

I am studying $p$-adic Rings and let me explain my understanding and doubt here. As I understood, Let $p$ be a rational prime and $Z$ denotes ring of integers, then form cartesian product $$P=Z/pZ ...
2
votes
0answers
28 views

Counting tamely ramified Galois extensions of $\mathbb{Q}_p$ with a given Galois group.

For a homework exercise, I'm to determine for each $p$ the number of non-isomorphic tamely ramified Galois extensions $K/\mathbb{Q}_p$ such that $\operatorname{Gal}(K/\mathbb{Q}_p) \cong ...