Exercise 3.3.15 (Eisenstein's criterion)

Question

Use Eisenstein's criterion to show that for every
$n \ge 1$, there is an irreducible polynomial over $\mathbb{Q}$ of degree $n$.

Hassaan Naeem · Accepted Answer

extbf{Solution:}
Let $f(t) = a_0 + ... + a_nt^n \in \mathbb{Q}[t]$ with $n \geq 1$.
For $n \geq 1$, we can always choose an $f \in \mathbb{Q}[t]$ such that $f(t)=a_nt^n + a_0$, and we can further
always choose an $a_n, a_0$ and $p$ such that $p 
mid a_n,\ p|a_0,\ p^2 
mid a_0$. Hence, we have $f(t) = a_nt^n + a_0$ fulfilling the
Eisenstein criterion, and hence $f(t)$ is irreducible over $\mathbb{Q}$. As an example, we can always choose
$f(t) = t^n +2$ and $p=2$.

Exercise 3.3.15 (Eisenstein's criterion)

Answers

Comments

Exercise 3.3.15 (Eisenstein's criterion)

Answers

Comments

Add answer