4 votes

Is the $p$-adic norm continuous?

It is not. If it were continuous, it would be continuous at $0$. Then, there would exist $\delta>0$ such that if $|x|<\delta$ then $|||x||_p- ||0||_p|=||x||_p<1$. However, this is impossible ...
4 votes
Accepted

How to introduce $p$-adic numbers to undergraduates?

Several years ago, I directed an informal undergraduate reading course on $p$-adic numbers. We followed Neal Koblitz's book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions (GTM volume 58). I ...
  • 4,385
4 votes
Accepted

elementary proof of $\log(-1)=0$ in $\mathbb{Q}_2$

Expanding @Merosity's idea, here is a solution: Let $N$ be a positive integer. Then \begin{align*} 0 = \frac{1 - (1 - 2)^{2^N}}{2^N} &= \sum_{k=1}^{\infty} \frac{1}{2^N} \binom{2^N}{k} (-1)^{k-1} ...
3 votes
Accepted

Is $\mathbb{Z}_p$ the only discrete valuation ring of $\mathbb{Q}_p$

No, neither statement is true. $\mathbb{Q}_p$ contains $\mathbb{Q}$, so it contains the localizations $\mathbb{Z}_{(q)} = \mathbb{Q} \cap \mathbb{Z}_q$ (given by the rational numbers whose denominator ...
3 votes
Accepted

How many unique, prime binary sequences are there of length n, modulo rotations?

The standard terminology is that a word is aperiodic if it doesn't consist of repetitions of another word. Given an aperiodic word we can consider its cyclic rotations, and among all of them there is ...
2 votes

Hasse Invariant for Integral Quadratic Forms

Given a quadratic form $q(u)=b(u,u)$ with $b$ a symmetric bilinear form $U\to \Bbb{Q}_p$ where $U$ is a free $\Bbb{Z}_p$-module (the $p$-adic integers) of rank $r$ and $p\ne 2$. Take $w\in U$ such ...
  • 70.6k
1 vote
Accepted

Rational subsets in rigid analytic and adic spaces

Your equality $$\bigcup_n r(V_n)=\{x\in r(X): 1<|f(x)|\}$$ is incorrect in general. The reason basically is that you can (and will) have higher rank points $x\in r(X)$ with the property that $1<|...
  • 49.5k

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