I'm in an algebra (ring) class and I'm looking at a previous midterm (I have attached it here to prove that this is not homework problem). Can anyone tell me how to answer question 3 and 5? I will repeat again in here: 3) Let R = Z[sqrt(2)] and P = <sqrt(2)> the ideal generated by sqrt(2) a) Prove that P is a prime ideal. b) Let D = Rp (localization of R at complement of P, aka ring of fractions of R w.r.t. (R-P) ) Let Pp be ideal of Rp generated by sqrt(2). Is Pp a prime ideal of Rp? Prove answer. 5) Let R = ZxZ (direct product of integer sets with operations defined as usual componentwise). Let I = < (4,9), (6,12) > ideal generated by those two elements. How many elements (cosets of I) does R/I have? Any help would be appreciated. Let me repeat that this is not homework, as I have attached file proving that these are past test questions (which I got from school's math society website).