Book recommendations (abstract algebra and number theory)

[AfterSunShine]

Hi,
For an engineer who graduated and finished typical Cal A,B,C + Linear Algebra + ODE, what book do you recommend to start reading to be a transition to advanced pure math subjects like abstract algebra and number theory?

I did deep google search & concluded that that book supposed to include logic, introduction to set theory & maybe review of number sets (integers, rational ... etc).
I think those are the subjects needed to be included in that "needed" book to start building good foundation in order to get deep in pure mathematics.

Can you suggest a book?
I will be glad if you can give short sequence of subjects to be learned, something like logic ---> intro to set theory ---> number theory ---> abstract algebra. i.e. sequence of learning.

Thanks

 

[The Bill]

May suggestion for a short sequence would be:

1. Any decent discrete mathematics textbook. Preferably a non-current edition, since they can be purchased cheaply. I usually recommend Rosen 4e or thereabouts, but there are many good options. That covers all the logic and set theory minimum prerequisites, and these books are typically very reader-friendly. I can personally vouch for the second through fifth editions of Rosen in that regard. Rosen also includes a little basic number theory which should help your intuition for later on.

My advice is to do a lot of the proof exercises in whatever discrete mathematics text you pick. This course is often a mathematics or theoretical computer science student's first class where they are expected to work multiple proofs on every exam.


After that, it's up to you whether to tackle algebra first or work through some more number theory material that doesn't presuppose any abstract algebra experience.

If you want to jump straight into some number theory at this point, Number Theory by George E. Andrews is quite nice. It's also a Dover book. It also makes a good alternate source for helping to understand some basic results alongside later texts. Neal Koblitz's text A Course in Number Theory and Cryptography also has few prerequisites, but requires more mathematical maturity and is more expensive. But, if you want to really learn the math behind lots of modern cryptography, you'll be technically prepared for it at this point.

For abstract algebra, there are many decent introductory texts. I usually recommend Pinter's A Book of Abstract Algebra, Dover edition. It is very reader friendly, and as a Dover book, quite economical. It covers all the topics one really needs to cover in a two semester undergraduate course on algebra, and a bit more besides. I can't think of any books that have pedagogy on par with Pinter that can be readily purchased at less than five times what it costs.

A nice next step is A Primer of Analytic Number Theory by Stopple. It's an undergraduate introduction to topics like the Prime Number Theorem and the Riemann zeta function. It also covers a fair bit of traditional algebraic/arithmetic number theory.

For more number theory, I can't think of a better next step than A Classical Introduction to Modern Number Theory by Ireland and Rosen (a different Rosen than above.) Others will probably have more recommendations. It's not cheap, but my best alternative recommendations would be combinations of textbooks adding up to a similar or greater total.

That about covers my first thoughts on what you asked for.

 

[fresh_42]

Your request is not very specific. You should consider consulting lecture notes. You can't go wrong with this basic stuff you listed. Search for "abstract algebra + pdf" or "group theory + pdf". The "+pdf" part will almost automatically lead you to lecture notes from some universities. Other basics would be topology, commutative algebra, and number theory.

 

[mathwonk]

try these free notes:
https://www.math.uga.edu/sites/default/files/inline-files/4000.01-05.pdf

if you like them, try others from the collection of math 4000 notes here:

https://www.math.uga.edu/directory/people/roy-smith

 

[malawi_glenn]

Try one of the "Introductory mathematics" here:
https://www.physicsforums.com/threa...-online-math-books-and-lecture-notes.1044710/

Then you could do some formal real analysis and linear algebra, like "understanding analysis" by Abbott, and "linear algebra done right" by Axler. After that, you could do some more formal multivariable analysis/differential geometry and abstract algebra and topology.

 

[AfterSunShine]

@TheBill
Thank you for your detailed reply. Appreciated.

@fresh_24
Thank you for your reply. My request is specific. Basically a book considered as an introduction to advanced mathematics. In other words, a book considered as introduction to abstract math including logic, proof methods, introductory set theory & few topics in number sets.
I really don't like these pdfs, I prefer book with chapters & exercises at end of each section/chapter.

@mathwonk
Thank you for your reply. Appreciated.

@malawi
Thank you your reply. It is exactly what I need.

If anyone has other recommendation, please mention.