Consider the ring R = Z[i], the ring of Gaussian integers, and the polynomial ring R[x]. a) Prove that R[x] is a Euclidean domain. Provide a detailed Euclidean function and demonstrate the Euclidean algorithm within R[x]. b) Determine whether R[x] is a Principal Ideal Domain (PID) and justify your answer. c) Investigate the irreducibility of the polynomial f(x) = x² + 1 in R[x]. Provide a comprehensive proof of its reducibility or irreducibility. d) Assuming f(x) is reducible, factorize it into irreducible polynomials in R[x]. If it is irreducible, explain the implications for the ring R[x]/(f(x)).

18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .

If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .

Let Q denote the field of rational numbers, R the field of real numbers, and C the field of complex. Determine whether each of the following polynomials is irreducible over each of the indicated fields, and state all the zeroes in each of the fields. a. x22 over Q, R, and C b. x2+1 over Q, R, and C c. x2+x2 over Q, R, and C d. x2+2x+2 over Q, R, and C e. x2+x+2 over Z3, Z5, and Z7 f. x2+2x+2 over Z3, Z5, and Z7 g. x3x2+2x+2 over Z3, Z5, and Z7 h. x4+2x2+1 over Z3, Z5, and Z7