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.

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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 $x \in \mathbb{Z}_2^{\star}, y ...
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$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, ...
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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: $\left|\sum_0^N (p-1)p^i+1 \right|_p ...
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$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 ...
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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 ...
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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 ...
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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 ...
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25 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? ...
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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 ...
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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 ...
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26 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 ...
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29 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 \{ ...
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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 $\mathbb{Q}$ ...
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44 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 ...
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39 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. ...
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27 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 : ...
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46 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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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$0$ 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 $\bigcap_{n \in \mathbb{N}_0 p^n \mathbb{Z}_p}=0$ ...
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110 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 ...
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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: ...
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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 $\mathbb{Z}_p \ ...
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$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 $GL_2(\mathbb{Q}_p)$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Visualizing Balls in Ultrametric Spaces

I've been reading about ultrametric spaces, and I find that a lot of the results about balls in ultrametric spaces are very counter-intuitive. For example: if two balls share a common point, then one ...
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$t$-adic topology (on $\mathbb F_p(1/t)$)

Recently I found this interesting discussion about algebraically closed fields of positive characteristic. In the answer marked as the top answer, I read about the $t$-adic topology. The $t$-adic ...
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Understanding the mechanics of P-adic topologies

I am trying to work out how it is that we actually work open sets on a p-adic topological space and how I would relate it to open sets in a point set topology. According wiki here: We have that open ...
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100 views

Proving existence of $\overline{\Bbb Q_p}$ without AC

The proof that every field has an algebraic closure is known to require at least a weak form of AC, the boolean prime ideal theorem. But I recall reading somewhere that for concrete, sufficiently ...
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Determine if $\sum_{q=1}^{\lceil n/2\rceil}R_q(n)$ gives the number of divisors of $n$.

Let $$R_q(n)=\left\{\begin{array}{lll} r\left(\dfrac n{2q-1}\right)&\text{if }(2q-1)\mid n\\ 0&\text{otherwise}\end{array}\right\},$$ where $r(n)$ is the ruler function, i.e., the $2$-adic ...
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191 views

Every Cauchy sequence converges

SENTENCE: The p-adic numbers are complete with respect to the p-norm, ie every Cauchy sequence converges. PROOF: Let $(x_i)_{i \in \mathbb{N}}$ a Cauchy-sequence in $\mathbb{Q}_p$. We want to show ...
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$\sum_{\zeta^p=1}(\zeta-1)^n$

Given $n\geq0$ let $$ z_n=\sum_{\zeta^p=1}(\zeta-1)^n $$ where $p$ is an odd prime number (summation extended to all $p$-th roots of 1). It is clear that: $z_n\in\Bbb Z$ (it's a Galois invariant sum ...
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28 views

Extending a DVR could produce not a DVR

I'm reading Tate's paper about $p$-divisible groups. In Chapter $(2.4)$ he asserts that if you take $R$ a complete DVR with residue field $k$ of characteristic $p>0$, $K$ its field of fractions, ...
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sequence $\{a^{p^{n}}\}$ converges in the p-adic numbers.

Let $a\in \mathbb{Z}$ be relatively prime to $p$ prime. Then show that the seqeunce $\{a^{p^{n}}\}$ converges in the $p$-adic numbers. This to me seems very counter intuitive. Since $(a,p)=1$ the ...
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246 views

At which p-adic fields does the equation have no solution?

I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't have, I have to find at which p-adic fields it has ...
3
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1answer
90 views

Why doesn't the equation have a solution in $\mathbb{Q}_2$?

I have to find for which primes $p$, the equation $x^2+y^2=3z^2$ has a rational point in $\mathbb{Q}_p$. According to my notes: Obviously, $\forall p \in \mathbb{P}, p \nmid 2 \cdot 3$, there is a ...
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35 views

How do we solve this congruence?

I am looking at the proof of Hensel's Lemma. Hensel's Lemma: Let $F(x)=a_0+a_1x+ \dots+ a_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a p-adic $\alpha_1 \in \mathbb{Z}$ such that: ...
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What does $p\mathbb{Z}_p$ mean?

I am looking at Hensel's Lemma: Let $F(x)=a_0+a_1x+ \dots + a_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p$-adic number ($p>2$) $\alpha_1 \in \mathbb{Z}_p$, such that: $$F(\alpha_1) ...
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34 views

convergence of the sequence $10^{-n}$ in the p-adic numbers

Let $p$ be prime. I am tasked to prove that the sequence $10^{-n}$ does not converge in $\mathbb{Q}_{p}$ for any $p$ where $\mathbb{Q}_{p}$ is the set of p-adic numbers. For $p=2$ or $5$, we see ...