Thread: The Maximal ideals of R[x]

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 ?

Originally Posted by CropDuster

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 .

Thanks. I think I see what you're saying, but what does IR[x] denote? And, p(x) is irreducible in IR[x] right? And if so:



right?

Originally Posted by CropDuster

Thanks. I think I see what you're saying, but what does IR[x] denote?

Denotes

And, p(x) is irreducible in IR[x] right? A:

I proposed a polynomial of degree 3. It is easy to generalize.

Originally Posted by FernandoRevilla

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 .

So what if p(x) is irreducible and has degree 2?

Originally Posted by CropDuster

So what if p(x) is irreducible and has degree 2?

Use that in a principal ideal domain every nonzero prime ideal is maximal.