91Ó°ÊÓ

A Latin square \(L\) is called self-orthogonal if \(L\) and its transpose \(L\) "form an orthogonal pair. a) Show that there is no \(3 \times 3\) self-orthogonal Latin square. b) Give an example of a \(4 \times 4\) Latin square that is self-orthogonal. c) If \(L=\left(a_{i j}\right)\) is an \(n \times n\) self-orthogonal Latin square, prove that the elements \(a_{i i}\), for \(1 \leq i \leq n .\) must all be distinct.

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

Does the Division Algorithm (for polynomials) hold in the integral domain Z \(\\{x]\) ? Explain.

\text { Construct a finite field of } 27 \text { elements. }