Reddit mentions of Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften (322))

The knowledge required will take you most, if not all the way, through your Ph. D. dissertation in number theory. Focus on getting admitted to a good graduate school and finding a good advisor, but in the meantime you might want to try reading Hartshorne and Neukirch.

These are likely to be the hardest math books you've ever read, but you have to start somewhere! (though I might throw in Atiyah-Macdonald for some more commutative algebra background for Hartshorne)

There are a lot of canonical texts on both subjects. One of the classics on ant is Neukirch.

You should definitely find a book which has a good coverage of transcendence degree and transcendence basis. It's a simple concept but really enlightening.