[2105.13216] Galois module structure of the units modulo $p^m$ of cyclic extensi

Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^...

p adic number theory - Example of degree $n$ ramified, but not totally ramified

I'm looking for an example of degree $n$ ramified but not totally ramified example over $\Bbb Q_p$. I can find degree $n$ ramified extension, for example, $\Bbb Q_p({p^{1/n}})/\Bbb Q_p$. $p^{1/n}$'s

linear algebra - Let $\mathcal{P}_n$ denote the space of polynomials of degree $

Question: Let $\mathcal{P}_n$ denote the space of polynomials of degree $n$. $$\mathcal{P}_n = \{ p_0 + p_1 x + p_2x^2 + \cdots p_nx^n\}.$$ Define $\mathcal{L}:\mathcal{P}_n \to \mathcal{P}_{n-1}$ by

functions - How to find $p(n+2)$ for a $n$th degree polynomial? - Mathematics St

Here's the exact question: If $p(x)$ denotes a polynomial of degree $n$, such that $p(k) = 1/k$ for $k = 1, 2 , 3 , ..., n + 1$, determine $p(2019)$ for $n = 2017$ Here was my initial approach: Tak...

linear algebra - Showing that $\dim_FF[x]/\langle p \rangle=n$. - Mathematics St

Given $F$ is a field and $p$ is a polynomial over $F$ with degree $n$. We let $\langle p\rangle$ be the principal ideal of all polynomials over $F$ which are multiples of $p$. I want to show that $\

Finding an irreducible polynomial of degree $n$ in $\mathbb Q[X]$ with real root

Context. I was trying to prove that for a given $n$, there exist a totally real number field of degree $n$. I understood that it was equivalent to find a polynomial $P$ such that (i) $P\in\mathb...

algebraic number theory - Norm $\text{N}_{L/ \Bbb Q_p}(\pi_L)$ of Uniformizer of

Let $ K/ \Bbb Q_p$ a finite extension of $p$-adic field $\Bbb Q_p$ of degree $[K:\Bbb Q_p]=n$ and let $ \pi_L$ be a uniformizer of $L$. Question: Can we say something interesting / "distinguis...

group theory - Can a degree $p^n$ field extension always be factored in a sequen

Suppose $L/K$ is a field extension of degree $p^n$ for some prime $p$ (if necessary, assume the characteristic of $K$ is not $p$). Then, is it always possible to find a sequence of extensions $K =...

calculus - let $p(x)=a_0 +a_1x ++a_nx^n \ $ is real polynomial of odd degr

let $p(x)=a_0 +a_1x +......+a_nx^n \ $ is real polynomial of odd degree such that $a_0a_n <0$. show that $p$ has at least two real roots. i know it will have at least one root by using IVP prop...

algebra precalculus - A polynomial $P$ of degree $2015$ satisfies the equation $

A polynomial $P$ of degree $2015$ satisfies the equation $P(n) = \frac{1}{n^2}$ for $n = 1, 2, \dots, 2016$. Find $2017P(2017)$. My try : Let $$x^2P(x) - 1 = (x-1)(x-2)\cdots(x-2016)(ax+b).$$ Sinc...