Is an Algebraically Closed Integral Domain Always a Field?

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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
 
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What property of a field does an integral domain lack? How does being algebraically closed fill that gap?
 
"Algebraically closed" is "overkill". You really only need a small result that follows from algebraically closed.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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