91Ó°ÊÓ

Construct the affine plane \(A P\left(\mathbf{Z}_{3}\right)\). Determine its parallel classes and the corresponding Latin squares for the classes of finite nonzero slope.

Use the Euclidean algorithm for polynomials to find the ged of each pair of polynomials, over the designated field \(F\). Then write the gcd as \(s(x) f(x)+t(x) g(x)\), where \(s(x), t(x) \in F[x]\). a) \(f(x)=x^{2}+x-2, g(x)=x^{5}-x^{4}+x^{3}+x^{2}-\) \(x-1\) in \(\mathbf{Q}[x]\) b) \(f(x)=x^{4}+x^{3}+1, g(x)=x^{2}+x+1\) in \(\mathbf{Z}_{2}[x]\) c) \(f(x)=x^{4}+2 x^{2}+2 x+2, g(x)=2 x^{3}+2 x^{2}+\) \(x+1\) in \(\mathbf{Z}_{3}[x]\)

Construct a finite field of 27 elements.