# Thread: The Maximal ideals of R[x]

1. ## 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 $\displaystyle \mathbb{R}[x]$ 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 $\displaystyle \mathbb{R}[x]$?

2. Originally Posted by CropDuster
What I am confused about is: why can't a polynomial with degree >2, generate a maximal ideal in $\displaystyle \mathbb{R}[x]$?

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 .

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?

4. 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.

5. 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?

6. 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.