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J. Neukirch Book (Algebraic Number Theory), page $50$ It has been mentioned that, The Legendre symbol $\left( \frac{a}{p} \right)=+1 \ \text{or} \ -1$ according as $x^2 \equiv a \mod p$ has solution …

Consider $\Bbb Q_3$ and its quadratic extension $\Bbb Q_3(\sqrt{3})$. I want to find the ramification index $e$. We have the valuation groups as follows: For $\Bbb Q_3$, the valuation group $G=\{|a| …

Let $F$ be a finite algebraic extension of $\mathbb{Q}_p$. Let $K$ is an abelian extension of $F$ of degree $n$, then show that $$ [F^{\times}: N_{K/F}(K^{\times})]=n,$$ where $N_{K/F}$ denotes the no …

If $K$ is a finite extension of $\mathbb{Q}_p$ degree $n$, index of ramification $e$, and residue field degree $f$, and if $\pi$ chosen so that $\text{ord}_p \pi=\frac{1}{e}$, then every $\alpha \in …

Let $K \supset \mathbb{Q}_p$ be the $p$-adic field with ring of integers $\mathcal{O}_K$ and $\pi_K$ be its uniformizer . Let $L$ be an unramified extension of $K$ of dgree $d$ and ring of integers $\ …

Let us consider the $p$-adic field $\mathbb{Q}_p$. Let $\pi$ be the uniformizer of $\mathbb{Q}_p$. Consider the extension $K_i$ of $\mathbb{Q}_p$ by adjoining $(p^i-1)^{th}$ roots of $\pi$, i.e., $K_ …

I got a lecture from the link-https://www.math.uni-bonn.de/people/ja/thecurve/the_curve_lecture_15_1_2020.pdf. Unfortunately, there are previous lectures of this series which I could not find. Can y …

Let us consider the $p$-adic field $\mathbb{Q}_p$. Let $\pi$ be the uniformizer of $\mathbb{Q}_p$. Consider the finite extensions $K_i$ of $\mathbb{Q}_p$ by adjoining $(p^i-1)^\text{th}$ roots of $\p …

Why is it necessary that when we are defining a filtration of algebraic objects there must be a topology associated to the filtration? For example, a descending filtration of group has the topology w …

I think you don't need the reverse inclusion. Note that $\pi^r\mathcal{O}_K=\{x \in K~|~ \text{ord}_p(x) \geq r\}$. Thus, $U_r=1+\pi^r \mathcal{O}_K=1+\{x \in K~|~ \text{ord}_p(x) \geq r\}$. Now note …

Consider the irreducible polynomial $h(t)=t^n+c_{n-1}t^{n-1}+\cdots +c_1t+p \in \mathbb{Z}_p[t]$. Consider the ring of $p$-adic integers and maximal ideal $p \mathbb{Z}_p$. I want to show all the coef …

Consider the $p$-adic field $K \supset \mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and maximal ideal $\mathfrak{m}_K$ whose $n^{th}$ higher unit group is defined by $$ U^{(n)}=\{u \in \mathcal …

To find the Weierstrass factorisation of the polynomial $f(x)=p-x+px^2$, we solve for the roots $ \alpha=\frac{1 \pm \sqrt{1-4p^2}}{2p}$. Now the square roots converges in $p$-adic ring integer $\math …

In the p-adic Hodge-Tate theory, what does the $\phi$-function in the definition of $(\phi, N)$-module mean? I mean what is the definition of $\phi$-function here? I got it that, A $\phi-$ modu …

Let $K \supset \mathbb{Q}_p$ be the $p$-adic field. Let $O$ be its ring of integers. Let us denote by $O_{(p)}$ be the localisation of $O$ at the prime/maximal ideal $(p)$. Consider the another ring $ …