Question: (15 = 8+7 marks) Let w = 2217/3 = -1+iV3, let R = = ) =1+1v3, let R = Z[w]. Let p be a positive

(15 = 8+7 marks) Let w = 2217/3 = -1+iV3, let

(15 = 8+7 marks) Let w = 2217/3 = -1+iV3, let R = = ) =1+1v3, let R = Z[w]. Let p be a positive prime integer which is not equal to 3. (a) Prove that R is an Euclidean domain, and find all the units of R. (b) Prove that the ideal pR is a maximal ideal of R, if and only if, p = -1 (modulo 3). (15 = 8+7 marks) Let w = 2217/3 = -1+iV3, let R = = ) =1+1v3, let R = Z[w]. Let p be a positive prime integer which is not equal to 3. (a) Prove that R is an Euclidean domain, and find all the units of R. (b) Prove that the ideal pR is a maximal ideal of R, if and only if, p = -1 (modulo 3)

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