91Ó°ÊÓ

Show that each of the following polynomials is irreducible over \(\mathbb{Q}\) : $$ \begin{array}{r} 3 x^{4}-8 x^{3}+6 x^{2}-4 x+6 ; \quad \frac{2}{3} x^{5}+\frac{1}{2} x^{4}-2 x^{2}+\frac{1}{2} ; \\ \frac{1}{5} x^{4}-\frac{1}{3} x^{3}-\frac{2}{3} x+1 ; \quad \frac{1}{2} x^{4}+\frac{4}{3} x^{3}-\frac{2}{3} x^{2}+1 \end{array} $$

Let \(F\) be any field. Explain why, if \(a(x)\) is a quadratic or cubic polynomial in \(F[x]\), \(a(x)\) is irreducible in \(F[x]\) iff \(a(x)\) has no roots in \(F\).

If \(a(x)\) and \(b(x)\) have the same roots in \(F\), are they necessarily associates? Explain.