I am trying to determine the maximal ideals in the following rings:
By reasoning as follows:
An ideal $I$ is maximal iff the quotient $R[X]/I$ is a field and by using the fact that in a PID, if $a$ is irreducible, then the ideal $(a)\triangleleft R$ is maximal.
Then I am stuck because there are many irreducibles, could anyone give me some hints of each case or some clear directions of how to proceed? Or if there are any intuitive or clever methods?