I think this time I didn't misread the question (hurray! 't was about freaking time), so if I did a mistake it was a "honest" one...and I did: the ideal is NOT the same as the ideal for some prime ; the latter is the ideal of all pol's all whose coef's are divisible by the prime , whereas the former (the one the OP asked about) is , which indeed is a maximal ideal. I confused these two.
Anyway, reading the PDF file I attached to muy first post the OP, hopefully, could have realized the above and overcome my misdirections.
Tonio