Prove or disprove that {Z109, +, x} is a Galois Field?

*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.

Let R = F_2[X], the ring of polynomials with coefficients from the field with two elements. Let J be the ideal generated by X^3+1, so specifically it is the set of all polynomials divisible by X^3+1. What is the product (X^2+J).(X^2+J) in the factor ring R/J? a. X b. 0+J C. 0 d. X+J e. Not sure

0 1 2 6 -18. 0 9 1. Given that F = 1 1 A = (FTF)-¹ FTY. 2. Having computed A from #1 above: Y = 0 compute I. Write down the equation of the best fit line (i.e. the estimation function), noting that the first entry of A is the intercept and the second, the slope. II. Compute Ŷ = FA, i.e. where the data point should have been to lie exactly on the line. 3. Next, compute the least squared error 2 E = ||Y – Y|| NOTE: All computations must be done by hand. Show detailed steps taken.