91Ó°ÊÓ

Compute the sum \(f(x)+g(x)\) and product \(f(x) \cdot g(x)\) for the given polynomials \(f(x)\) and \(g(x)\) in the given polynomial ring \(R[x]\). $$ \begin{array}{lll} 2 x^{2}+x+1 & \text { and } 3 x^{2}+2 & \text { in } \mathbb{Z}_{6}[x] \end{array} $$

Factor the indicated polynomial \(f(x)\) completely into irreducible factors in the polynomial ring \(F[x]\) for the indicated field \(F\). $$ f(x)=x^{4}+1 \quad F=\mathbb{Z}_{2} $$

Factor the indicated polynomial \(f(x)\) completely into irreducible factors in the polynomial ring \(F[x]\) for the indicated field \(F\). $$ f(x)=x^{4}+4 \quad F=\mathbb{Z}_{5} $$

Find all zeros of the indicated \(f(x)\) in the indicated field. $$ f(x)=x^{3}+x^{2}+x+1 \quad \text { in } \mathbb{C} $$

In Exercises 1 through 7 (a) Show that the indicated set \(I\) is an ideal in \(Q[x] .(b)\) Find a polynomial \(g(x)\) in \(Q[x]\) such that \(I=\langle g(x)\rangle(c)\) Determine whether \(I\) is a maximal ideal in \(\mathrm{Q}[x]\). \(I=\\{f(x) \in \mathbb{Q}[x] \mid f(\sqrt{2})=0\\}\)

Construct a field with the indicated number \(n\) of elements. $$ n=16 $$

Describe the indicated quotient rings. $$ \mathbb{R}[x] /\left\langle x^{2}+x\right\rangle $$

Calculate the quotients and remainders on division of the indicated \(f(x)\) by the indicated \(g(x)\) in the indicated polynomial rings \(F[x]\). $$ f(x)=x^{4}+x^{3}+x^{2}+x+1 \quad g(x)=x+1 $$

Find all zeros of the indicated \(f(x)\) in the indicated field. $$ \begin{aligned} &f(x)=x^{8}-1\\\ &\text { in } \mathbb{R} \end{aligned} $$

Calculate the quotients and remainders on division of the indicated \(f(x)\) by the indicated \(g(x)\) in the indicated polynomial rings \(F[x]\). $$ f(x)=x^{5}+5 x^{3}+3 x^{2}+2 \quad g(x)=x^{2}+4 x+5 \quad \mathbb{Z}_{7}[x] $$