4. (10 points) Let k be a field and k[x, y, z] kl3]. Define the quotient ring R= k[x, y, z]/(x2 (y2 + 1)). (a) Show that R is an integral domain and find the field of fractions K = frac(R). (b) Show that R is a normal ring. Hint: Show that R= k(x)[y] n k(z)[y] in K. (c) Show that R is a UFD if and only if y2 + 1 is irreducible over k. Hint: Show that y2 + 1 is irreducible iff x is prime. Use Nagata's Criterion for one direction; for the other direction show x is irreducible. (d) The coordinate ring of the sphere S2 is defined by: A = R(x, y, z]/(x2 + y2 + z2 1) Show that CR A = C[x,y,z]/(xz (y2 + 1)) and conclude that CR A is not a UFD. Note: A is a UFD but this is harder to show. 4. (10 points) Let k be a field and k[x, y, z] kl3]. Define the quotient ring R= k[x, y, z]/(x2 (y2 + 1)). (a) Show that R is an integral domain and find the field of fractions K = frac(R). (b) Show that R is a normal ring. Hint: Show that R= k(x)[y] n k(z)[y] in K. (c) Show that R is a UFD if and only if y2 + 1 is irreducible over k. Hint: Show that y2 + 1 is irreducible iff x is prime. Use Nagata's Criterion for one direction; for the other direction show x is irreducible. (d) The coordinate ring of the sphere S2 is defined by: A = R(x, y, z]/(x2 + y2 + z2 1) Show that CR A = C[x,y,z]/(xz (y2 + 1)) and conclude that CR A is not a UFD. Note: A is a UFD but this is harder to show