<p>                               Let <img alt="$\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$" height="38" src="https://latex.artofproblemsolving.com/a/5/2/a522f1b44474cef3df06a8a745fec43bde20be0f.png" width="186" /> where <img alt="$i = \sqrt{-1}.$" height="18" src="https://latex.artofproblemsolving.com/4/a/2/4a2e92550cefae6e71b66ea8a86ac40ebcd6b9af.png" width="72" /> Find the value of the product</p> <p><img alt="\[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\]" height="51" src="https://latex.artofproblemsolving.com/9/2/6/926a1478b07a9cae8e5bf2a5d40181beea580ea6.png" style="float: right;" width="150" /></p>

Question No: 448

Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product

$\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).$

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Tags: Algebra

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