TITLE: What are some major open problems in Galois theory?
QUESTION [7 upvotes]: Few days back one of my friend and I were discussing about Galois life and his ideas. Though we are not trained in Galois theory, but I am recently started to learn it by myself and hope to take up research on it some times soon. So I am wondering about the open problems in Galois theory? Is there any? Please add a reference to it. Thanks.

REPLY [15 votes]: One of the most active problems in Galois theory is the so called "Inverse Galois Problem" concerning whether or not every finite group appears as the Galois group of some extension of the rational numbers. It is a problem not only concerning Galois theory but also High Level Finite Group theory. This is an old problem but it is still unsolved.
To have a brief introduction to this subject the Wikipedia's article is pretty easy to understand: https://en.wikipedia.org/wiki/Inverse_Galois_problem
Hope It helped.