Solve Abstract Algebra math problem

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I'm associate professor in Kyiv Politechnic Institute, have been giving lectures on discrete math, including the beginning of abstract algebra. I could try solve your task; however, time is very tight. PS Can we use the well-known theorem: factorizing an integral domain by maximal ideal gives a field. If we can, the last task become easy. PPS A new multiplication can obviously defined as ordinary polynomial multiplication with mod by x^2+1. BTW, ideal (x^2+1) is maximal in R[x], so R[x]/(x^2+1) is a field.

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