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## Hot answers tagged p-adic-number-theory

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### Why does an argument similiar to 0.999...=1 show 999...=-1?

If you want to understand the mathematics behind these things, it is all based upon the notions of 'convergence' and of 'limits'. If you read any first course in analysis textbook you will find the ...
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114 votes

### Why does an argument similiar to 0.999...=1 show 999...=-1?

In the $10$-adic numbers it is true that $\dots 9999 = -1$. More precisely, the series $\sum_{n=0}^{\infty} 9 \cdot 10^n$ converges in $\mathbb{Q}_{10}$ and its limit there is $-1$.
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### Why does an argument similiar to 0.999...=1 show 999...=-1?

As other users have noted, it doesn't make sense to have infinitely many nines to the left of the decimal point. This is because the sequence $9, 99, 999, \ldots$ doesn't converge to anything, unlike ...
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33 votes
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### Units of p-adic integers

This must be covered in almost every text on the $p$-adic numbers; I think the book of Gouvêa is the best of these. And your statement is not quite true: for $p=2$, $\mu_{p-1}$ is trivial all right, ...
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26 votes
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### Ring of the integer $p$-adic numbers $\mathbb{Z}_p$

Let's look at a different ring first: $R=k[[X]]$, the ring of formal power series over a field. We can also think of $R$ as the ring of sequences $f_0, f_1, f_2,\ldots$, where each $f_i$ is a ...
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25 votes
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### Method of finding a p-adic expansion to a rational number

The short answer is, long division. Say you want to find the $5$-adic expansion of $1/17$. You start by writing $$\frac{1}{17}=k+5q$$ with $k \in \{0,1,2,3,4\}$ and $q$ a $5$-adic integer (that is, ...
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23 votes

### Why does an argument similiar to 0.999...=1 show 999...=-1?

One thing to think about is how this plays with modular arithmetic, if you're familiar. Basically, arithmetic mod $10^n$ is arithmetic where we only care about the last $n$ digits of a number and is ...
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23 votes

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14 votes
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The question of whether two "numbers" are "equal" is a somewhat subtle one. For example, lets work with a simpler number, namely "2". Certainly $2\in\mathbb{Z}$, but also $2\in\mathbb{Q},2\in\mathbb{... • 11.9k 14 votes ### Why are$p$-adic numbers and$p$-adic integers only defined for$p$prime? I was going on at too great length in a comment to a discussion between Henning Makholm and Hurkyl, so let me put it all into an answer instead: To show that$\Bbb Z_{10}\cong\Bbb Z_2\times\Bbb Z_5$, ... • 60.5k 13 votes Accepted ### sequence$\{a^{p^{n}}\}$converges in the p-adic numbers. Recall that the Euler totient function has values$\phi(p^n)=p^{n-1}(p-1)=p^n-p^{n-1}$for all$n$. This means that for all$a$coprime to$p$we have the congruence $$a^{p^n}\equiv a^{p^{n-1}}\pmod{... • 125k 13 votes ### Method of finding a p-adic expansion to a rational number In the case of -1/6, it’s very easy:$$-\frac{1}{6} = \frac{1}{1-7} = \sum_{k=0}^∞ 7^k.$$What happens: In the p-adic numbers, the sequence p^k is a null sequence (as |p^k|_p = p^{-k} \overset{... • 17.9k 13 votes Accepted ### Totally ramified extensions of \mathbb{Q}_p This is an exercise I’ve never done, but it should be a lot of fun. What is the general Eisenstein polynomial in this case? it’ll be$$ X^3 + 2aX^2+2bX+2(1+2c)\,, $$where a, b, and c can be any ... • 60.5k 13 votes Accepted ### A puzzle involving 10-adic numbers I saw this observation in a math book once when I was 16 or so and was totally baffled at the time. It's nice to know I understand it now! As you say, the starting point is to use CRT, which allows us ... • 387k 12 votes Accepted ### The maximal unramified extension of a local field may not be complete This is a natural question, because it’s really easy to get overwhelmed by the situation. In the case of the completion of the maximal unramified of a local field k, here’s the way that I look at ... • 60.5k 12 votes ### If B is an abelian group, then is B{\otimes}_{\mathbb Z}{\mathbb Z}_p isomorphic to {\varprojlim}B/p^{n}B? This is false in general: consider B = \mathbb{Q}. Then \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}_p \cong \mathbb{Q}_p, but \mathbb{Q} / p^n \mathbb{Q} = 0 for every n, so \varprojlim_n \... • 2,472 12 votes Accepted ### Various p-adic integrals Not really an full answer, but some comments (that hopefully answer some of your queries). There seems to be a big confusion here : what do we want to integrate, i.e. to define \int_{\mathbb{Z}_p} f(... • 2,034 12 votes ### Why are p-adic numbers and p-adic integers only defined for p prime? You can speak about 10-adic numbers just fine, but they don't behave as nicely as p-adic numbers. For example, the 10-adic integers have zero divisors, so no matter how you complete or massage ... 11 votes Accepted ### examples of unramified extensions of \mathbb{Q}_p You get unramified extensions of \Bbb Q_p by adjoining roots of unity of order prime to p; alternatively, by adjoining (p^n-1)-th roots of unity. The finite unramified extensions of \Bbb Q_p ... • 60.5k 11 votes ### Why introduce the p-adic numbers? On one hand, the p-adic numbers are extremely natural objects of study: by Ostrowski's theorem every nontrivial absolute value on \mathbf Q is equivalent to either the usual absolute value or the ... • 496 11 votes Accepted ### Classical number theoretic applications of the p-adic numbers One of my favourite classical results using p-adic methods in elementary number theory is the theorem of Skolem-Mahler-Lech: This is a theorem about linear recurrence sequences, which are sequences ... • 4,315 10 votes ### Which p-adic fields contain these numbers? There are several ways of attacking questions like this. The most general is to transform your defining equation (such as for \sqrt{-7}, which I’ll use as the type example) into something to which ... • 60.5k 10 votes Accepted ### Is 123456788910111121314\cdots a p-adic integer? This is not an n-adic integer or a p-adic integer. A pivotal idea of a p-adic integer is that it can be well-represented in a sort-of-base-p,$$ x = \sum_{k \geq \ell}a_kp^k,$$and in ... • 87.6k 10 votes ### Why does an argument similiar to 0.999...=1 show 999...=-1? This argument is similar to the one$$ \sum_{n=1}^\infty n = -1/12$$which went viral a few years ago. You are actually using methods which were originally designed for manipulating absolutly ... • 659 10 votes Accepted ### Is there concept of continuous curve and surfaces in p-adic field? The$p$-adic numbers are a metric space, and thus a topological space. Therefore continuity is well defined, either by the$\varepsilon,\delta\$ definition that you know from Calculus, or by the ...
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