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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2008 • Question 9

Question

Math 6310 • Algebra Fall 2008 • Question 9

Take a regular \(n\)-sided polygon inscribed in a circle of radius 1. Label the vertices \(P_1,\ldots ,P_n\), and let \(\lambda _k\) be the length of the line joining \(P_k\) and \(P_n\) for \(1\leqslant k\leqslant n-1\). Prove that \[ \lambda _1\cdots \lambda _{n-1}=n. \]

Raw LaTeX

Take a regular \(n\)-sided polygon inscribed in a circle of radius 1. Label the vertices
\(P_1,\ldots,P_n\), and let \(\lambda_k\) be the length of the line joining \(P_k\) and \(P_n\) for
\(1\leqslant k\leqslant n-1\). Prove that
\[
\lambda_1\cdots\lambda_{n-1}=n.
\]

Topic Tags

Cyclotomic Fields