Intereting Posts

Triangle Inequality $\sum_{\text{cyc}}\frac{a}{b+c}<2$
Reflexive normed spaces are Banach
Why is $L^{1}(G)$ unital if and only if $G$ is discrete?
What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$?
How many triangles are there in the picture?
Pappus Chain Recursive Radii
The sequence $\{a_k\}_{k=1}^\infty$ is increasing. What about $\{b_k\}_{k=1}^\infty$?
Does the fact that matrices $A, B$ are similar imply that $A+cI$ is similar to $B+cI$?
Is Cantor's Diagonal Argument Dependent on Tertium Non Datur
multiplication in H-space and loopspace of the H-space
Prove that $a_n=1+\frac{1}{1!} + \frac{1}{2!} +…+ \frac{1}{n!}$ converges using the Cauchy criterion
Show $4 \cos^2{\frac{\pi}{5}} – 2 \cos{\frac{\pi}{5}} -1 = 0$
How prove this $\lim_{x\to+\infty}(f'(x)+f(x))=l$
Global sections of $\mathcal{O}(-1)$ and $\mathcal{O}(1)$, understanding structure sheaves and twisting.
$\sqrt{13a^2+b^2}$ and $\sqrt{a^2+13b^2}$ cannot be simultaneously rational

I am looking at some notes that give an example of a Cauchy sequence that doesn’t converge in $\mathbb{Q}$ with respect to the $p$-adic absolute value. Their example is to let $1 < a< p-1$ and consider the sequence $(x_n) = (a^{p^n}).$ They claim that the sequence doesn’t converge in $\mathbb{Q}$. Their argument is that if the sequence had a limit $x \in \mathbb{Q}$, then we would have $x^p = x$ (which I understand) and $|x-a|_p < 1$ (which I also understand). But then they claim that therefore $a$ is a non-trivial $(p-1)$-th root of unity. I do not understand that. Could someone please clarify?

- “Number of Decompositions into $k$ Powers of $p$”-Counting Functions
- Integral points on an elliptic curve
- Show that ${(F_n^2+F_{n+1}^2+F_{n+2}^2)^2\over F_{n}^4+F_{n+1}^4+F_{n+2}^4}=2$
- Prove that the class number of $\mathbb{Z}$ is $1$
- Powers of $x$ as members of Galois Field and their representation as remainders
- Let $d$ be any positive integer not equal to $2, 5,$ or $ 13$ , then $\exists a, b \in \{2, 5, 13, d\}$ such that $ab − 1$ is not a perfect square?
- Primes and arithmetic progressions
- Understanding the trivial primality test
- How to prove $\gcd(a,\gcd(b, c)) = \gcd(\gcd(a, b), c)$?
- Prove or disprove : $a^3\mid b^2 \Rightarrow a\mid b$

Suppose $(x_n)=(a^{p^n})$ converges to $x$ in $\mathbb{Q}$ with respect $p$-adic norm. Then

using strong triangle inequality and little fermat’s theorem it is easy to see $|x-a|_p<1$.

It is easy to prove $(x_n) $ is Cauchy sequence. Also $x=\lim_{n\rightarrow\infty}x_n=\lim_{n\rightarrow\infty}x_{n+1}=\lim_{n\rightarrow\infty}x_n^p=(\lim_{n\rightarrow\infty}x_n)^p=x^p.$ We proved $|x-a|_p<1$ and $x^{p-1}=1$. Since $|x-a|_p<1, $ implies that $p|x-a$. Since $x^{p-1}=x$ and $x\in\mathbb{Q},$ implies that $x=1$. But $x$ cannot be 1. Suppose $x=1$. Then $p|(1-a)$. That is $1-a=pk$ for some $k\in\mathbb{Z}$, that is $a=pk+1$. If $k>0$ then $a>p$ and if $k<0$ then $a<1$, which is a contradiction, since $1<a<p-1.$ Hence $(x_n)$ not converges to $x$ in $\mathbb{Q}$ with $p$-adic norm.

- What does $GL_n(R)$ look like?
- How to find B-Spline represenation of an Akima spline?
- Visual references for the Riemann-Stieltjes integral.
- Composite functions and one to one
- Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$
- Sufficient condition for a *-homomorphism between C*-algebras being isometric
- Can we prove that odd and even numbers alternate without using induction?
- Prove that a series converges if $|a_n|<1$
- Show that when $BA = I$, the solution of $Ax=b$ is unique
- Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$
- How do you sketch the Poisson distribution function: $L(\mu;n)=\frac{\mu^{n}e^{-\mu}}{n!}$?
- Entropy Solution of the Burger's Equation
- group of order 30
- example of a nonempty subset is closed under scalar multiplication but not a subspace
- Proving that the terms of the sequence $(nx-\lfloor nx \rfloor)$ is dense in $$.