Is an Algebraically Closed Integral Domain Always a Field?

[Gabbey]

Homework Statement

Let R be an integral domain and algebraically closed. Prove it follows that R is a field.

The Attempt at a Solution

I guess it follows from the definitions but I can't specify which it is

 

[Dick]

What property of a field does an integral domain lack? How does being algebraically closed fill that gap?

 

[HallsofIvy]

"Algebraically closed" is "overkill". You really only need a small result that follows from algebraically closed.