Don't worry about what fields are "hot" or not (point 5). For one thing, that can change very quickly when you least expect it. New results or sudden applications may catapult a particular area to the fore, and answers to questions can doom a field to obscurity. It's going to take you several years from now until you are done, and there is no guarantee that what is "hot" or "on the rise" right now will still be hot or on the rise when you are done.

I would also advice to give preponderance to 1 and 2. If you become a mathematician, you will spend your time thinking about these things, and they better be things that you *want* to spend time thinking about. If doing your research becomes a chore, you are doomed, no matter how good you are at it, or how hot the field is. You'll be looking for excuses *not* to do it.

Point 4 is not really something that can go into deciding what field to study, because you sort of need to know a bit more about the field to really be able to answer it.

Mainly, in my opinion, you'll want to do something that *you* personally find exciting, interesting, and fun, for whatever reason. Sure, it's nice for the people who are working in hot areas and can easily get grants, since it is always better to be rich and healthy than sick and poor, but remember: if things go well, you'll be doing this for 30-50 years. You want to enjoy it.

If you have already learnt group theory, I may suggest you to go through
the book 'theory of algebraic numbers' by Pollard & Diamond. It's a really
good treatise to start off. You don't even need to know the definition of
ring to read this book. Everything is given there in a very well setup.
After having finished that book, you may pay a look at 'A Classical
Introduction to Modern Number Theory' by Ireland & Rosen.

## Best Answer

It is very rare that an entire field of math can be dismissed as "easy and remarkably elementary". In all fields, there are easy questions, but they have been solved and we have moved on to the difficult questions.

Basicgraph theory could perhaps be compared to basic calculus, but there's plenty more to say about graph theory.(It's conceivable that we run out of questions about a topic, because there was no depth to it. It could be argued that Euclidean geometry has reached this state. But there are certainly plenty of questions in graph theory that we still don't know the answer to.)

Now, where areas of math differ greatly is the amount of work you have to do

beforeyou get to ask a question. In graph theory, there are plenty of problems that can be asked with only the most basic of definitions - problems that you could explain to a stranger on a plane when asked "so what is it that you do?" Someone working in K-theory would have a hard time even getting to the point where a question could be formulated.I personally find this feature attractive: I feel that a problem which is easy to state and hard to solve needs no justification to be considered interesting, whereas a problem that needs half an hour of theory to even pose is less obviously natural. And the complicated stuff is always there: that's where we get the tools to solve the simple problems. But this is a personal preference, and other mathematicians quite reasonably disagree.