A finite field, F, of order 53 is a vector space of dimension 3 over Z5. Suppose a is a generator of F. Let M be any 3×3 matrix with entries from Z5 (careful here, the elements are from the field of scalars, not from the vector space). If we convert an element x∈F to a vector (relative to the basis {1,a,a2}), then we can multiply it by M (with M on the left) to create another vector, which we can translate to a linear combination of the basis elements, and hence another element of F. This function is a vector space homomorphism, better known as a linear transformation (implemented with a matrix representation relative to the basis {1,a,a2}). Notice that each part below becomes less general and more specific.
(a) Create a non-invertible matrix R and give examples to show that the mapping described by R is a vector space homomorphism of F into F.
(b) Create an invertible matrix M. The mapping will now be an invertible homomorphism. Determine the inverse function and give examples to verify its properties.
(c) Since a is a generator of the field, the mapping a↦a5 can be extended to a vector space homomorphism (i.e. a linear transformation). Find a matrix M which effects this linear transformation, and from this, determine that the homomorphism is invertible.
(d) None of the previous three parts applies to properties of multiplication in the field. However, the mapping from the third part also preserves multiplication in the field, though a proof of this may not be obvious right now. So we are saying this mapping is a field automorphism, preserving both addition and multiplication. Give a nontrivial example of the multiplication-preserving properties of this mapping. (This is the Frobenius map which will be discussed further in Chapter 21.)