proof writing - Showing a list of polynomials is linearly dependent in $\mathcal{P}_m(\mathbb{F})$

Note: In case the notation isn't common, $\mathcal{P}_m(\mathbb{F})$ denotes the set of all polynomials $\mathbb{F} \to \mathbb{F}$ with at most degree $m$. I'm having trouble with the following

complex analysis - P_n be a sequence of polynomials of degree at most m whic

Definition: A mapping $P : E \rightarrow F$ is said to be a polynomial of degree at most m if it can be represented as a sum $$ P = P_0 + P_1 + \ldots + P_m $$ where $P_j \in P_a(E;F)$. Question: ...

J.D./M.P.P.M. - Admissions - University of Maine School of Law

Maine Law’s J.D./M.P.P.M. dual degree program with the University of Southern Maine's Muskie School of Public Service is an accelerated program that allows students to obtain a Juris Doctor (J.D.)...

vector spaces - Find $m$ degree $q^{m-1}$ polynomials which give a bijection $\m

Let $q$ some prime power. Now take the field $\mathbb{F}_{q^m}$. We need to find polynomials $p_1(x),p_2(x),\dots,p_m(x)\in \mathbb{F}_{q^m}[x]$ such that $\deg{p_i(x)}=q^{m-1}$ and they also sati...

number theory - Why $[\mathbb{Q}(\sqrt[n]{p}, \sqrt[m]{q}):\mathbb{Q}] = nm$?

As title says, I want to prove that $\mathbb{Q}(\sqrt[n]{p}, \sqrt[m]{q})$ is degree $nm$ extension of $\mathbb{Q}$ when $p \neq q$ are distinct primes. By Eisenstein's criterion, $x^{n} - p$ is

number theory - The minimal polynomial of a primitive $p^{m}$-th root of unity o

Proposition 7.13 of Neukirch's ANT states that for a primitive $p^{m}$-th root of unity $\zeta$ ($p$ prime) the extension $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is totally ramified of degree $(p-1)...

field theory - Let $f(x)\in \Bbb{F}_p[x]$ be an irreducible polynomial of degree

Let $m,n$ be positive integers and $p$ be a prime. Consider the finite field $\Bbb{F}_{p^n}$ and let $f(x)\in \Bbb{F}_p[x]$ be an irreducible polynomial of degree $m$. Let the factorisation of $f(x...

complex analysis - To prove $\left|\frac{p_n(z)q_m(z)}\right|\leq \frac{M|z|

To prove there exist $M>0$ and $a_0>0$ such that for $|z|>a_0$, $$\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$$ where $p_n$ and $q_m$ are the polynomials of degree $n$ and $m$

abstract algebra - Let $p_j\in K[X]$ a polynomial of degree $j$. It is true that

Let $p_j\in K[X]$ a polynomial of degree $j$. It is true that an arbitrary collection $p_0,p_1,...,p_m$ is a basis of $K_m[X]$? Here $K$ is a field and $K_m[X]:=\{p\in K[X]:\deg(p)\le m\}$ is a ve...

How existence of an irreducible polynomial in $F_p(x)$ of degree $n$ guarantees

I read somewhere that if $\pi$ is an irreducible polynomial of degree $m$ then $F_p(x)\ \backslash \left< \pi \right>$ is a finite field of order $p$. What is $F_p(x)\ \backslash \left< \pi \