  We establish new explicit zero-free regions for the Dedekind zeta-function.
Two key elements of our proof are a non-negative, even, trigonometric
polynomial and explicit upper bounds for the explicit formula of the so-called
differenced logarithmic derivative of the Dedekind zeta-function. The
improvements we establish over the last result of this kind come from two
sources. First, our computations use a polynomial which has been optimised by
simulated annealing for a similar problem. Second, we establish sharper upper
bounds for the aforementioned explicit formula.
