**Question:**

Let $$f(X)=(X-x_{1})(X-x_{2})\cdots (X-x_{n})$$ be an irreducible polynomial over the field of rational numbers,with integer coefficients and real zeros.

Prove that $$\prod_{1\le i<j\le n}|x_{i}-x_{j}|\ge\dfrac{n^n}{n!}$$

This result is fell interesting, and This problem is from problem book, and this problem is name Siegel created it, and I searched sometimes and can't find it.and can't solve this problem, can someone help me?

The Queen of Mathematicsin which you will find the same result. See Chapter 22 pp. 461-462 (here) in which the result is discussed as a special case of Minkowski's proof of a conjecture due to Kronecker. $\endgroup$ – Benjamin Dickman Dec 23 '14 at 6:19