3
votes
2answers
33 views

Build field extension and solve equation

Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field. As I understand we need to build $\mathbb{F}_{5^{2}}$. Field $\mathbb{F}_5$ contains ...
1
vote
1answer
61 views

Fields of polynomials . Proving that a belongs to k as a root

if $f(x)\in k[x]$, where $k$ is a field, then $a\in k$ is a root of $f(x)$ iff $x-a$ divides $f(x)$ in $k[x]$. My result ... If $a$ is a root of $f(x)=q(x)(x-q)$ and if we let $f(x)=q(x)(x-a)$,then ...
7
votes
4answers
378 views

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
1
vote
2answers
187 views

Irreducible polynomials with integer coefficients over Q

Suppose p(x) is an irreducible polynomial over Q of degree n, with integer coefficients. If p(x) has two roots r1 and r2 satisfying r1r2 = 5, prove that n is even. Attempt at solution: Because the ...
2
votes
2answers
306 views

Proof existence of field extension of $\mathbb{F}_p$ containing the $r$-th primitive root of unity

I have to show the following: Let $p$ be a prime and $r \in \mathbb{N}$ with $\gcd(r,p)=1$. Prove the existence of a field extension $E$ of $\mathbb{F}_p$ which contains an $r$-th primitive root ...
38
votes
5answers
3k views

Continuity of the roots of a polynomial in terms of its coefficients

It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
8
votes
3answers
1k views

How to solve polynomial equations in a field and/or in a ring?

I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples: (1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ...