I plan to study algebraic number theory (arithmetic and Galois theory) in order to start research in cryptography and coding theory and all the staff... Also, i would like to introduce myself to algebraic geometry having in view the weil conjectures and its connections with number theory. Please, can you give me an estimation of the necessary time in order to be ready for doing serious research in these fioftlds ?
Thanks !!!
If you are just starting to study Galois theory and algebraic number theory, there is really no guarantee that you will ever get to the point of doing "serious" original research in those fields. Mathematical talent may be there or not.
On the other hand, if your goal is to understand a.n.t and Galois well enough to apply them to cryptography, then I would say that one year of serious study, perhaps 18 months, should be enough. I say this because of you were to study these in courses at a good university, it would encompass two or three semesters at most to get a good grasp of the basics; and that is while having other course work as well.
