Rationalizing fractions over finite fields

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To rationalize the expression 1/(1 + w^k) in a finite field, one must recognize that w is a primitive n-th root of unity, which means it belongs to a finite field rather than the complex numbers. The challenge lies in manipulating the finite field properties to eliminate the denominator. Unlike complex numbers, operations in finite fields require different techniques, such as finding multiplicative inverses or utilizing field characteristics. The discussion emphasizes the need for a clear understanding of finite field arithmetic to approach the problem effectively. Rationalizing fractions in this context involves leveraging the unique properties of finite fields.
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Homework Statement


Let w be a primitive n-th root of unity in some finite field. Let 0 < k < n. My question is how to rationalize

[\tex]\dfrac{1}{1 + w^k}[\tex].

That is, can we get rid of the denominator somehow? I know what to do in the case of complex numbers but here I'm at a loss. Thanks!

Homework Equations


The Attempt at a Solution

 
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Not sure what is the command for latex but I meant 1/(1 + w^k).
 
Isn't (1 + w^k) a complex number?
 
It isn't. It's in a finite field.
 

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