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...

number theory - Is $F times/N_{L/F}(L times)$ non trivial or infinite when $L/

Let $L/F$ be an extension of global fields of degree $n\geq 2$, not necessarily Galois. I thought it was well known that the group $F^\times/N_{L/F}(L^\times)$ is always non trivial, and even infin...

abstract algebra - Prove $[L:\mathbb{F_p}]=n$ - Mathematics Stack Exchange

Let $f(x) \in \mathbb{F_p}[x] $ be an irreducible polynomial of degree $n$. Let $L$ be the splitting field of $f$. Prove $[L:\mathbb{F_p}]=n$. If $a_1,...,a_n$ are the roots of $f(x)$, then $L=\...

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

real analysis - $P$ is a monic polynomial of degree $n$ , then which are correct

Suppose that $P$ is a monic polynomial of degree $n$ in one variable with real coefficients and $K$ is a real number. Then which of the following statements are necessarily correct ? If $n$...

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

real analysis - $\lim_{x\to\infty}\dfrac{P(x)}{Q(x)} = \lim_{x\to-\infty}\dfrac{

This is a true statement, right? And $L = 0$. I could prove this using algebraic limit theorems and taking $P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ and $Q(x) = b_m x^m + b_{m-1} x^{m-1} + ......

Victoria Goulbourne | My fave type of session, I N T E R V A L S 🫶🏻🏃🏼♀️ mayb

231 likes, 5 comments - victoria_goulbourne - June 26, 2024: "My fave type of session, I N T E R V A L S 🫶🏻🏃🏼♀️ maybe not in the 25 degree heat, I had sunburn and resembled a beetroot 👍🏻 #run...

Does a polynomial $P(x)$ of degree $n$ over $\mathbb{F_q}$ which satisfy $P(x)|x

Does a polynomial $P(x)$ of degree $n$ over $\mathbb{F_q}$ which satisfies $$P(x)|x^{l}-1$$ only for $l\ge q^n-1$ is irreducible? If it's, how it can proven?

Implication of dot product $(Q_n(x), 1)\equiv \int_a^b {p(x)Q_n(x)dx} = 0$ in $L

Dot product $(Q_n(x), 1)\equiv \int_a^b {p(x)Q_n(x)dx} = 0$ in $L_2[a,b,p]$, where $Q_n(x)$ is a polynomial of $n$-th degree. According to my textbook this happens only if $Q_n(x)$ has at least one