Let $\| x \| =\sqrt{x^T x}$ be the Euclidean norm on $\mathbb{R}^n$. Consider the point $z \in \mathbb{R}^n$, and the plane $P = \{x \in \mathbb{R}^n : a^T x = b\}$ where $0 \neq a \in \mathbb{R}^n$, $b \in \mathbb{R}$. Orthogonal projection gives the point \begin{align}\label{1}\tag{1} y = z – \frac{(a^T z – b)}{a^T a} […]

Suppose we have a number field $K$ with ring of integers $\mathcal O_K$. Let $\frak p$ be a prime ideal in $K$ lying over $p\in \mathbb Q$. Then, using the $\frak p$-adic norm, we may define the $p$-adic ring of integers $\mathbb Z_{\frak p} = \{ x\in K: |x|_{\frak p}\leq 1\}$. I know that $\mathcal […]

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 topology is unfamiliar to me and I could not seem to find anything via google. I do know something about the $p$-adic topology though (cf this […]

I was reading this article (http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf) to see some applications of $p$-adic numbers outside mathematics, and came across these two identities: $\sum_{n=0}^\infty (-1)^n n!(n+2) = 1$ and $\sum_{n=0}^\infty (-1)^n n!(n^2-5) = -3$. Since no $p$ was stated, I figured there must be some trick that shows these are the values in any $\mathbb{Q}_p$, but I […]

Let $f \in \mathbb{Z}[x]$ be a separable monic polynomial, with $f(0) \neq 0$, and $p$ be a prime number. Also, let $L$ be the splitting field of $f$ over $\mathbb{Q}_p$ and let $a_1, \ldots, a_n \in L$ be all the roots of $f$. Finally, let $b_1, \ldots, b_n \in \mathbb{C}$ also be the roots of […]

How to prove that there are seven non isomorphic quadratic field extensions of $\mathbb{Q}_2$? Approach: I already proved that an unit in $\mathbb{Z}_2$ is congruent with $1,3,5,7 \mod 8$. (maybe that’s useful)

We know that every rational number can be written as a $p$-adic integer with expansion $\sum\limits_{n=-m}^\infty a_n p^n$, where $a_n\in\{0,\dots,p-1\}$ and $m\in\mathbb{N}$; therefore there exists an injection $\mathbb{Q} \hookrightarrow \mathbb{Q}_p$. But how do I show that $\mathbb{Q}_p$ is bigger? How do I find an example of a $p$-adic number which is not rational? I heard […]

Let $p$ be a prime number and $d$ a positive integer. If $n\in \mathbb{Z}_p$ (the ring of $p$-adic integers) how would one define $d^n$? Under what conditions would this be an element of $\mathbb{Z}_p$? Any help would be very much appreciated.

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 of the balls is contained in the other. The reason I find these results so counter-intuitive is that I can easily […]

Let $K$ be a finite extension of $\mathbb Q_p$. How to prove that if an element of $K$ has non negative valuation then it is algebraic over $\mathbb Z_p$? I would like also a reference for this proof (e.g. is it somewhere in Algebraic Number Theory by Neukirch ?).

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