  We show how to construct all the extensions of left braces by ideals with
trivial structure. This is useful to find new examples of left braces. But, to
do so, we must know the basic blocks for extensions: the left braces with no
ideals except the trivial and the total ideal, called simple left braces. In
this article, we present the first non-trivial examples of finite simple left
braces. To explicitly construct them, we define the matched product of two left
braces, which is a useful method to recover a finite left brace from its Sylow
subgroups.
