91Ó°ÊÓ

Find \(q(x)\) and \(r(x)\) as described by the division algorithm so that \(f(x)=g(x) q(x)+r(x)\) with \(r(x)=0\) or of degree less than the degree of \(g(x)\). $$ f(x)=x^{6}+3 x^{5}+4 x^{2}-3 x+2 \text { and } g(x)=x^{2}+2 x-3 \text { in } Z_{7}[x] $$

In Exercises 1 through 4 , find \(q(x)\) and \(r(x)\) as described by the division algorithm so that \(f(x)=g(x) q(x)+r(x)\) with \(r(x)=0\) or of degree less than the degree of \(g(x)\). $$ f(x)=x^{6}+3 x^{5}+4 x^{2}-3 x+2 \text { and } g(x)=x^{2}+2 x-3 \text { in } Z_{7}[x] $$

Find \(q(x)\) and \(r(x)\) as described by the division algorithm so that \(f(x)=g(x) q(x)+r(x)\) with \(r(x)=0\) or of degree less than the degree of \(g(x)\). $$ f(x)=x^{6}+3 x^{5}+4 x^{2}-3 x+2 \text { and } g(x)=3 x^{2}+2 x-3 \text { in } Z_{7}[x] . $$

In Exercises 1 through 4 , find \(q(x)\) and \(r(x)\) as described by the division algorithm so that \(f(x)=g(x) q(x)+r(x)\) with \(r(x)=0\) or of degree less than the degree of \(g(x)\). $$ f(x)=x^{6}+3 x^{5}+4 x^{2}-3 x+2 \text { and } g(x)=3 x^{2}+2 x-3 \text { in } \mathbb{Z}_{7}[x] $$

In Exercises 1 through 4 , find \(q(x)\) and \(r(x)\) as described by the division algorithm so that \(f(x)=g(x) q(x)+r(x)\) with \(r(x)=0\) or of degree less than the degree of \(g(x)\). $$ f(x)=x^{5}-2 x^{4}+3 x-5 \text { and } g(x)=2 x+1 \text { in } Z_{11}[x] $$

Find all generators of the cyclic multiplicative group of units of the given finite field. (Review Corollary 6.16.) $$ \mathrm{Z}_{17} $$

The polynomial \(x^{4}+4\) can be factored into linear factors in \(\mathbb{Z}_{5}[x]\). Find this factorization,

The polynomial \(x^{3}+2 x^{2}+2 x+1\) can be factored into linear factors in \(Z_{2}[x]\). Find this factorization.

The polynomial \(2 x^{3}+3 x^{2}-7 x-5\) can be factored into linear factors in \(Z_{11}[x]\). Find this factorization.

Is \(x^{3}+2 x+3\) an irreducible polynomial of \(Z_{5}[x]\) ? Why? Express it as a product of irreducible polynomials of \(Z_{5}[x]\).