1. In the ring R=Z [√2], let M = {a+b√2|2|a}. Answer the following (a) Show M is an ideal of R (b) Show M is a maximal ideal of R, directly, using the definition of maximal ideal = (c) Now show M is maximal by showing F RM (that is, the quotuent ring R mod M) is a field as follows: Just show that F has only two elements (note √2 € M ) and hence must be the field of two elements

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