Question

(PRIME IDEALS IN ?[X]) Consider the ring ?[X]. (a) Consider the map ? : ?[X] ? ? given by f ? f(0), which is obviously a ring homomorphism. Show that ker ? = (X). (b) Use part (a) and the First Isomorphism Theorem to show that (X) is a prime ideal of ?[X] that is not maximal. (c) Show that (2, X) = {f ? ?[X] : f(0) is even}. (d) Consider the map ? : ?[X] ? ?? given by f ? f(0) mod 2 (where ?? = ?/2? is the field of two elements), which is also a ring homomorphism. Show that ker ? = (2, X). (e) Show that (2, X) is a maximal ideal of ?[X]. (f) Show that (2, X) is not a principal ideal of ?[X]. (Hint: Suppose to obtain a contradiction that (2, X) = (f) for some f ? ?[X]. Use part (c) to try and derive a contradiction.)

          (PRIME IDEALS IN ?[X]) Consider the ring ?[X].
(a) Consider the map ? : ?[X] ? ? given by f ? f(0), which is obviously a ring homomorphism. Show that ker ? = (X).
(b) Use part (a) and the First Isomorphism Theorem to show that (X) is a prime ideal of ?[X] that is not maximal.
(c) Show that (2, X) = {f ? ?[X] : f(0) is even}.
(d) Consider the map ? : ?[X] ? ?? given by f ? f(0) mod 2 (where ?? = ?/2? is the field of two elements), which is also a ring homomorphism. Show that ker ? = (2, X).
(e) Show that (2, X) is a maximal ideal of ?[X].
(f) Show that (2, X) is not a principal ideal of ?[X]. (Hint: Suppose to obtain a contradiction that (2, X) = (f) for some f ? ?[X]. Use part (c) to try and derive a contradiction.)

(PRIME IDEALS IN ?[X]) Consider the ring ?[X].
(a) Consider the map ? : ?[X] ? ? given by f ? f(0), which is obviously a ring homomorphism. Show that ker ? = (X).
(b) Use part (a) and the First Isomorphism Theorem to show that (X) is a prime ideal of ?[X] that is not maximal.
(c) Show that (2, X) = f ? ?[X] : f(0) is even.
(d) Consider the map ? : ?[X] ? ?? given by f ? f(0) mod 2 (where ?? = ?/2? is the field of two elements), which is also a ring homomorphism. Show that ker ? = (2, X).
(e) Show that (2, X) is a maximal ideal of ?[X].
(f) Show that (2, X) is not a principal ideal of ?[X]. (Hint: Suppose to obtain a contradiction that (2, X) = (f) for some f ? ?[X]. Use part (c) to try and derive a contradiction.)

Added by Hannah M.

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