$GF(29)^2$ is created by adjoining the root of the irreducible quadratic $p=x^2+7x+15$ to the field $GF(29)$ . The cubic polynomial $q=Y^3+(26x+26)Y^2+(8x+22)Y+13x+23$ is irreducible over this new field.
a. If the root of the given cubic is adjoined to $GF(29)^2$ in order to form a larger field extension, how many elements are in the new field?
b. Using the matrix representation find the product of the two polynomials $Y^2+x+1$ and $Y^2+xY+1$ I know how to make fields and how to generate companion matrix and do the multiplication of the given polynomials in part b.
But I am not sure that I am building the matrix correctly. I need help in understanding the question properly.
It is my assignment question I am not looking for a straight answer but for well build explanation which helps me understand the base concept.