8th Putnam 1948 Problem A5 Let α 1 , α 2 , ... , α n be the nth roots of unity. Find ∏ i<j (α i - α j ) 2 .
Let α k = exp(2 p i(k-1)/n). So α 1 = 1, and the other α k are roots of x n-1 + x n-2 + ... + x + 1 = 0. Hence (1 - α 2 ), ... , (1 - α n ) are roots of (1 - x) n-1 + ... + (1 - x) + 1 = 0. The coefficient of x 0 is n and the coefficient of x n-1 is (-1) n-1 . Hence ∏ 2 n (1 - α k ) = n. Hence n/α k = (α k - α 1 )(α k - α 2 ) ... (α k - α n ), where the product excludes (α k - α k ). Hence n n /(α 2 ...α n ) = ∏ i not equ j (α i - α j ) = (-1) n(n-1)/2 ∏ i<j (α i - α j ) 2 . But (α 2 ...α n ) = (-1) n-1 . Hence ∏ i<j (α i - α j ) 2 = (-1) (n-1)(n-2)/2 n n . 8th Putnam 1948 © John Scholes jscholes@kalva.demon.co.uk 11 Mar 2002