Solution

For a monic polynomial with roots

r_{1}, \dots, r_{n}

, its discriminant is

Disc (f) = \prod_{i < j} (r_{i} - r_{j})^{2} .

Here

Disc (g) = (u^{3} - v^{3})^{2}

. Both

u^{3} - v^{3}

and the Vandermonde product in the

α_{i}

are alternating homogeneous cubics. Evaluation at

(α_{1}, α_{2}, α_{3}) = (0, 1, - 1)

fixes the constant and gives

(u^{3} - v^{3})^{2} = - 27 \prod_{i < j} (α_{i} - α_{j})^{2} = - 27 Disc (f) .

Using

α_{1} + α_{2} + α_{3} = 0

\sum_{i < j} α_{i} α_{j} = a

, and

α_{1} α_{2} α_{3} = - b

, direct expansion gives

uv = - 3 a, u^{3} + v^{3} = - 27 b .

Consequently

g (X) = X^{2} + 27 b X - 27 a^{3},

and

Disc (g) = 729 b^{2} + 108 a^{3} = - 27 (- 4 a^{3} - 27 b^{2}) .

Thus

Disc (f) = - 4 a^{3} - 27 b^{2} .

Solved by gpt-5.6-sol high.

 Ancestors (11)