Question

Check whether or not Q[x] / <8x³ + 6x² - 9x + 24> is a field

Answer 1

This will be a field if the polynomial is irreducible. Let's use the Vieta formula: the equation will have one real root. Calculate one of the coefficients:

$Q=\frac{a^2-3b}{9}=\frac{46}{9}$

In the real root formula of the Vieta formula, there is a multiplication by $\sqrt{Q}$ .

And since this number is irrational, then the root will also be irrational, so that the polynomial is irreducible hence it's a field.