Let R = Z [x] and let I = be an ideal. 1) Prove that...

Transcribed Image Text:

Let R = Z [x] and let I = <3, x^2 +1> be an ideal. 1) Prove that R/I is isomorphic to Z3 [x] /<x^2 +1> ii) Conclude that R/I is a field and determine the number of its elements, show your steps iii) Prove or show a counterexample: Let S = Z3 [x] /<x^2+1>, then the class of x+1 is the multiplicative generator of S* (The Set of the Units of S) RING THEORY (Please prove explicitly all the results and state the theorems used) Let R = Z [x] and let I = <3, x^2 +1> be an ideal. 1) Prove that R/I is isomorphic to Z3 [x] /<x^2 +1> ii) Conclude that R/I is a field and determine the number of its elements, show your steps iii) Prove or show a counterexample: Let S = Z3 [x] /<x^2+1>, then the class of x+1 is the multiplicative generator of S* (The Set of the Units of S) RING THEORY (Please prove explicitly all the results and state the theorems used)

Answer rating: 100% (QA)

i To prove that RI is isomorphic to Z3x we need to find a bijective homomorphism from RI to Z3x and show that it preserves the structure of the ring D... View the full answer