This is not a reference request, that is, I have access to many textbooks I am happy with. What I don't know is, what are the things I need to know to get started?
My idea on the path of knowledge acquisition:
- Group theory
- Ring theory
- Quadratic number fields
- Fields
- Galois Theory
Is that how I should approach this? Have I missed anything? Should I study linear algebra rigorously first or is it irrelevant here?
Galois theory, as presented nowadays in undergraduate classes, is mostly linear algebra with group theory. A finite field extension $\mathbf{K}$ over $\mathbf{F}$, is a vector space over $\mathbf{F}$ and an automorphism of $\mathbf{K}$ over $\mathbf{F}$ is also a linear transformation from $\mathbf{K}$ to $\mathbf{K}$ as $\mathbf{F}$ vector space. So, I think linear algebra is kind of important here! For the ring theory part, you need to be more precise: after reviewing basic stuff in Ring theory, it's good to view some other things, not really neccessary for Galois theory, but might help to understand concepts better : Einsenstein Criterion, some quadratic rings over $\mathbf{Z}$, ($\mathbf{Z}[\sqrt5] \cdots$ )