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Jan 30, 2021 · Rather, one would have to investigate what "roles" the classical $\pi$ "plays" in the real numbers (e.g.: area and / or half the circumference of the unit circle; first positive zero of the sine function; four times the value of arctan at $1$; certain rational multiple of $\zeta$-values; certain multiple of the period of the complex logarithm ...

I'll give an answer that's more procedural. You can calculate the p-adic expansion for a given rational number using the following algorithm: Let a/b be a rational number, let p be a prime, and let k = 0, 1, 2 ... p - 1. Step 1. For all k, compute a2 = (a/b - k)*(b/p) ... 1st term is the k for which a2 is an integer. Step 2.

We find that the p-adic integers are formal neighborhoods of closed points in $\mathbb{Z}$, so they naturally show up in the deformation theory of arithmetic objects (Schemes over $\mathbb{Z}$) What are other reasons to consider p-adic integers/numbers?

Jun 5, 2018 · I want to know if the p-adic fields $\mathbb{Q}_p, \mathbb{Z}_p$ are characteristic 0 fields, or what are their characteristics? Actually, is $\mathbb{Z}_p$ a field at all? (note that the only definition i have seen and somewhat understood so far of the p-adic number fields is that they are extensions of $\mathbb{Q}, \mathbb{Z}$ respectively ...

Jun 19, 2015 · It can be shown (Ostrowski's theorem) that Q admits only 2 distances which are compatible with + and x : the "archimedaean" one, built from the usual absolute value, and the p-adic one, built from the "p-adic valuation" defined by v_p(a/b) = v_p(a) - v_p(b), where a, b are in Z and v_p (a) = the exact power of p dividing a (same thing for b).

Dec 16, 2020 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

In the recent literature on the history of p-adic numbers, I found two references that "suspect" that Gauss had something similar to p-adic analysis in his arsenal: The article " on the origin of p-adic analysis " by Peter Ulrich, which mentions Gauss's proof of his lemma on polynomials (from article 42 of the D.A) using valuation theory as ...

Jan 16, 2017 · Does the converse of Schur's lemma hold for smooth representations of finite length of p-adic group? Arguments for 128-bit processors now (or even a 96-bit) Rebbi Akiva's belief of the 10 tribes

Jun 30, 2022 · "P-adic Analysis compared with Real," by Svetlana Katok is a very gentle introduction to p-adic numbers. This text is suitable for an undergrad who has taken some analysis and topology. "A Course in p-adic analysis," by Alain Robert is a more terse and advanced book on the subject.

Dec 21, 2020 · I am a freshman and for a final project of a subject I have to give an introduction to p-adic numbers, I look for some sources (books, videos, articles) to be able to do my work, the only bases I have is a very basic "number theory" (construction of $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, through equivalence relations and the construction of ...