Find a maximal ideal of Z x Z If no such ideal exists then s

Find a maximal ideal of Z x Z. If no such ideal exists, then say so.(b)Find a prime ideal of Z x Z that is not maximal. If no such ideal exists, then say so.(c)Find a proper nonzero ideal in Z x Z that is not prime. If no such ideal exists, then say so.(d)Is Q[x]/(x62 - 6x + 8) a field? Justify your answer.

Solution

a) Z X pZ is maximal for any prime p.

b) Z X pZ is also prime ideal

c) pZ X qZ is not prime ideal

d) x^2 -6x + 8 = (x-4)(x-2) So its not prime ideal

it is not maximal