91Ó°ÊÓ

Let \(R\) be a ring with unity \(1,\) and let \(I_{1}, \ldots, I_{n}\) be ideals in \(R\) such that \(I_{i}+I_{j}=R\) for all \(i \neq j .\) Show that there exists a ring isomorphism: $$ \theta: R /\left(I_{1} \cap \ldots \cap l_{n}\right)=R / I_{1} \times \ldots \times R / I_{n} $$

Calculate \(\operatorname{gcd}(f(x), g(x))\) for the indicated \(f(x)\) and \(g(x)\) in the indicated polynomial rings \(F[x]\). Also, in each case find \(u(x)\) and \(v(x)\) such that \(\operatorname{gcd}(f(x), g(x))=u(x) f(x)+v(x) g(x)\) $$ f(x)=x^{3}+2 x+1 \quad g(x)=x+2 $$

In Exercises 8 through 14 determine whether the indicate ideals are maximal or not in Q \([x]\). Justify your answers. \(I=\left\langle x^{2}-5\right\rangle\)

Factor the indicated polynomial \(f(x)\) completely into irreducible factors in the polynomial ring \(F[x]\) for the indicated field \(F\). Show that \(f(x)=x^{3}+2 x+1\) is irreducible over \(\mathbb{Z}_{5}\)

Determine whether the indicate quotient rings are fields. Justify your answers. $$ \mathbb{Z}_{2}[x] /\left\langle x^{2}+x+1\right\rangle $$

Calculate \(\phi_{\alpha}(f(x))\) for the given fields \(E\) and \(F\). $$ \phi_{1}\left(x^{4}+1\right) \quad F=E=\mathbb{Z}_{2} $$

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. \(I=\left\langle x^{2}+x+1\right\rangle\)

Determine whether the indicate quotient rings are fields. Justify your answers. $$ \mathrm{Q}[x] /\left\langle x^{4}+1\right\rangle $$

Factor the indicated polynomial \(f(x)\) completely into irreducible factors in the polynomial ring \(F[x]\) for the indicated field \(F\). Show that \(f(x)=x^{3}+x+1\) is irreducible over \(\mathbb{Z}_{7}\)