So I am in the process of trying to do a proof, and I need show that the maximal ideals in are the irreducible polynomials of the form
What I am confused about is: why can't a polynomial with degree >2, generate a maximal ideal in ?
So I am in the process of trying to do a proof, and I need show that the maximal ideals in are the irreducible polynomials of the form
What I am confused about is: why can't a polynomial with degree >2, generate a maximal ideal in ?
For example, if p ( x ) in IR [ x ] has degree 3 then,
p ( x ) = ( x - a ) q ( x ) with q ( x ) in IR [x] and a in IR .
Then,
( p( x ) ) is contained in ( x - a ) and ( x - a ) is different from ( p ( x ) ) and IR [ x ] .
Edited: I forgot to say that p ( x ) had degree 3 .