There is considerable literature on this question, much of which can be revealed by doing a google search on "root separation". In particular, Wolfram MathWorld's "root separation" article is brief but to the point, and also see:

Polynomial Minimum Root Separation
George E. Collins
Department of Computer and Information Sciences, University of Delaware, Newark, DE 19716, U.S.A.
Received 21 June 2001; accepted 7 July 2001. Available online 28 February 2002.
Abstract
There is a well-known lower bound, due to Mignotte, for the minimum root separation of a squarefree integral polynomial, but no evidence for the sharpness of this bound. This paper provides massive computational evidence for a conjectured much larger bound, one that is approximately the square root of Mignotte’s bound.
References

(this is journal of symbolic computation).