# How to work out whether this factor ring is a field or not

• Mar 24th 2010, 10:01 AM
Siknature
How to work out whether this factor ring is a field or not
The ring R = Q[X,Y] where Q is the rational numbers

The ideal I = <X+1>

The factor ring is R/I = Q[X,Y]/<X+1>

How can i work out whether this factor ring is a field or not?
• Mar 24th 2010, 10:51 AM
tonio
Quote:

Originally Posted by Siknature
The ring R = Q[X,Y] where Q is the rational numbers

The ideal I = <X+1>

The factor ring is R/I = Q[X,Y]/<X+1>

How can i work out whether this factor ring is a field or not?

Theorem: if R is a unitary commutarive ring and I an ideal in R, R/I is a field iff I is a maximal ideal.

So is I= <X+1> maximal?

Tonio
• Mar 24th 2010, 11:11 AM
Siknature
Quote:

Originally Posted by tonio
So is I= <X+1> maximal?

Tonio

No, I=<X+1> is not maximal.

But that it is in fact what i was originally trying to prove and i thought i could use the theorem you have stated to do so.

What is a better way of proving that I=<X +1> is not maximal?