TITLE: Understanding $Gal(\bar k /k)$
QUESTION [5 upvotes]: According to this book, An Introduction to the Langlands Program,

One of the fundamental goals of modern number theory is to understand the Galois group $Gal(\bar k /k)$ where $k$ is a local or a global field.

What is meant by understanding this group?

REPLY [3 votes]: There is a big Mathoverflow discussion with the same title and 10 answers, some of them are quite informative depending on reader's background.  It begins just like this question,

I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G=$Gal$(\overline{\mathbb{Q}}/ℚ)$. What do people mean when they say this?

https://mathoverflow.net/questions/2791/