4. (10 points) Let k be a field and k[x, y, z] = k 131. Define...

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4. (10 points) Let k be a field and k[x, y, z] = k 131. Define the quotient ring R = k[x, Y, z]/(xz – (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(2)[y] in K. (c) Show that R is a UFD if and only if y? + 1 is irreducible over k. Hint: Show that y? + 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 C OR A = C[r,y, z]/(xz – (y² + 1)) and conclude that COR 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] = k 131. Define the quotient ring R = k[x, Y, z]/(xz – (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(2)[y] in K. (c) Show that R is a UFD if and only if y? + 1 is irreducible over k. Hint: Show that y? + 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 C OR A = C[r,y, z]/(xz – (y² + 1)) and conclude that COR A is not a UFD. Note: A is a UFD but this is harder to show.

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4 10 points Let k be a field and k x y z k l 3 Def ine the quot ient ring R k x y z x z y 1 ANS WER R f x y z x z y 1 k x y z f x y z k x y z a x z y 1 k x y z b y z y 1 k x y z c z z y 1 k x y z a b ... View the full answer