field theory - Why $\mathbb F_q/\mathbb F_p$ where $q=p^n$ is an extension of de

Let $q=p^n$ where $p$ is prime. Q1) Why is $X^{q}-X\in \mathbb F_p[X]$ irreducible ? Don't we have $X^q-X=X(X^{q-1}-1)$ ? Q2) Suppose it is irreducible, his degree is $p^n$, then $[\mathbb F_q:\...

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 $\

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=\...

field theory - Irreducible polynomial in $\mathbb{F}_p[x]$ divides $x^{p^n}-x$ -

Let $p(x)=x^{p^n}-x \in \mathbb{F}_p[x]$, $n \in \mathbb{N}$. I want to prove that any irreducible polynomial in $\mathbb{F}_p[x]$ with degree $n$ divides $p(x)$. Let $q(x)$ be this irreducible

calculus - Consider function f(x)=$\sqrt{x}$ and $p_n(x)$ as its Taylor polynomi

Consider function $f(x)=\sqrt{x}$ and $p_n(x)$ as its Taylor polynomial of degree $n$ about the point $x=9$.we want to compute the $E_n$ of $p_n(16)$(error). Can you prove as $n$ increases error

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?

functions - $f:P_{n}\rightarrow \mathbb{Z}^{n+1}$ is a bijection - Mathematics S

Let $P_{n}$ denote the set of polynomials of degree less than or equal to $n$ with integer coefficients. I want to show that $f:P_{n} \rightarrow \mathbb{Z}^{n+1}$ given by $f(a_{0}z^{n} + a_{1}z^{...

real analysis - What is the maximum number of points at which an $n$-degree poly

What is the maximum number $\phi(n)$ of roots of the function $e^{x^2}-p(x)$, where $p(x)$ is an $n$-degree polynomial. It is known that an $n$-degree polynomial intersects the function $e^x$ at a...

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 \

abstract algebra - Proving $\mathbb{F}_p[x]/\langle f(x)\rangle$ with $f(x)$ irr

I am attempting to prove what the title says, that $\mathbb{F}_p[x]/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements. I have already proven that for any fie...