I would appreciate suggestions for books to enhance my learning in algebra so as to be able to read Samuel's "Algebraic Theory of Numbers" and eventually at least begin Neukirch's "Algebraic Number Theory."
By way of background, I have gone through B. Gross's Harvard lectures on algebra several times. They were correlated to Artin, and included factoring and quadratic number fields, but did not cover modules or fields. Nor Galois Theory.
So I would like to get a good exposure to those areas that are particularly germane to ANT. (E.g. Artin does not have anything on perfect fields and only mentions algebraic closure in a short paragraph before the Fundamental Theory of Algebra.
Thanks very much.
Best Answer
I'll convert this to an answer, since it's getting somewhat long and I believe it answers the question.
Artin's book is more than enough preparation for Samuel's.
Maybe I can allay your fears about what Artin omits. In basic algebraic number theory your fields are either of characteristic zero (finite extensions of $\mathbf Q$ and their completions) or are finite fields (reductions of rings of integers modulo primes), and these are always perfect. Infinite extensions do not play a significant role until you start learning class field theory.
Note too that Samuel defines and proves basic properties about principal rings, modules, algebraic extensions, Galois extensions, and more; so in theory you wouldn't have to know much about those in order to begin reading. It's a very approachable book.