Let R= Z/5Z, the integers mod 5. The ring of Gaussian integers mod 5 is defined by R[i] = {a+ bi : a, b e Z/5Z and i = v-1}. Show that R[i] is not an integral domain (and hence not a field) by showing that 3+i is a zero-divisor in R[i].

*3.19 (R. A. Dean) Define F4 to be the set of all 2 x 2 matrices F4= F₁ = {[8 a+b] : a.b @P₂} . a, Prove that F4 is a commutative ring whose operations are matrix addition and matrix multiplication. (i) (ii) Prove that F4 is a field having exactly 4 elements. (iii) Show that I4 is not a field.

Prove that F2  is a field.

a) Calculate the splitting field K of x^3+2x^2-8x-8 over Q.  b) Calculate the degree of [K:Q].