It has been known for some time that to an algebraic field extension K of the rational numbers
one can associate a group of automorphisms of K which fix the rationals pointwise. Those groups
are called Galois groups (of K over Q) and they arise naturally from polynomials defined over Q.
One can then ask the following two questions: given a finite group G, is there an extension K of Q
such that G is its Galois group; what is the structure of the group of automorphisms of the
entire set of algebraic numbers which fix the rationals pointwise? The former is called ``the inverse Galois
problem'' and remains very much open, while the latter is a question about the so-called absolute
In this talk I shall construct the absolute Galois group and talk about Grothendiecks "Esquisse"
which is, as the name says, a sketch of a topological approach towards understanding an algebraical object.
I shall also say something about the inverse Galois problem over various base fields.