91Ó°ÊÓ

Write out the addition and multiplication tables for the following quotient rings. $$ \mathbb{Z}_{2}[x] /\left\langle x^{3}\right\rangle $$

In Exercises 8 through 14 determine whether the indicate ideals are maximal or not in Q \([x]\). Justify your answers. \(I=\left\langle x^{4}-1\right\rangle\)

Factor the indicated polynomial \(f(x)\) completely into irreducible factors in the polynomial ring \(F[x]\) for the indicated field \(F\). Show that \(f(x)=x^{4}-2\) is irreducible over \(Q\) but reducible over \(\mathbb{R}\).

Write out the addition and multiplication tables for the following quotient rings. $$ \mathbb{Z}_{3}[x] /\left\langle x^{2}\right\rangle $$

Find the \(n\) th roots of unity for the indicated \(n,\) and show that they form a cyclic subgroup of \(\mathbb{C}^{*}\) of order \(n .\) $$ n=4 $$

Factor the indicated polynomial \(f(x)\) completely into irreducible factors in the polynomial ring \(F[x]\) for the indicated field \(F\). Show that \(f(x)=x^{4}-2 x^{2}-4\) is irreducible over \(\mathbb{Q}\)

Find \(m(x)\) and \(n(x)\) in \(\mathbb{Z}_{5}[x]\) such that $$ x^{4}+4 x^{2}+3=m(x)\left(x^{2}+x+1\right)+n(x)(x+1) $$

Find the \(n\) th roots of unity for the indicated \(n,\) and show that they form a cyclic subgroup of \(\mathbb{C}^{*}\) of order \(n .\) $$ n=6 $$

Determine whether the indicate quotient rings are fields. Justify your answers. $$ Q[x] /\langle x-2\rangle $$

In Exercises 8 through 14 determine whether the indicate ideals are maximal or not in Q \([x]\). Justify your answers. \(I=\left\langle 6 x^{5}+14 x^{3}-21 x+42\right\rangle\)