I'm interested in learning Algebraic Number Theory and Galois Theory, so I was suggested Ireland Rosen and Stewart for them respectively. But the problem with the books is that the text is very very easy to understand and almost all the exercises are trivial, so I couldn't understand things properly from this.

Then I was suggested to work on Problems in Algebraic Number Theory by Murty, and I'm finding it to be very hard. Since it's supposed to be an "introductory first read" on ANT, does the fact I'm finding Murty hard suggest I lack the required mathematical maturity to solve the problems from it ?

Also what are some Galois Theory/Algebraic Number Theory somewhere between these two ? The exposition needn't be too hard to follow, but the exercises should at least take some time to solve.

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    $\begingroup$ For algebraic number theory, try Marcus’s “Number Fields”, from Springer’s Universitext series. I think you’ll find it more challenging than Ireland-Rosen. $\endgroup$ – Arturo Magidin Apr 5 at 5:55
  • $\begingroup$ Other suggestions are Milne's notes or the book by Neukirch. $\endgroup$ – asdq Apr 5 at 8:56
  • $\begingroup$ @asdq Thanks but the scary thing is that it as per Amazon review-er's it requires Grad level abstract algebra (I have undergrad level abstract algebra experience) expereince and morever one guy suggest to do the "simple Murty" book before doing this (and I find Murty to be hard...) $\endgroup$ – cdt Apr 5 at 9:01
  • $\begingroup$ Fair enough, I don't know the other books that have been mentioned. I suggest you have a look at the books and see for yourself, though. $\endgroup$ – asdq Apr 5 at 9:19

I might consider this set of notes written by Matt Baker for some algebraic number theory (minus Galois theory). The preface states

“Algebraic Number Theory is often presented in either a very elementary way which does not take full advantage of students’ backgrounds in Abstract Algebra, or in a very abstract and high-powered way which runs the danger of divorcing itself in the students’ minds from the concrete origins and applications of the subject. We have tried to steer a middle ground between these approaches.”

This seems like it might be a good fit for you. With regard to difficulty related to your knowledge of algebra, I would suggest you keep a copy of Dummit and Foote next to you while you read. If you don’t understand an argument, take a break and read a bit in D&F about the objects being used and then come back to whatever you end up reading.

(Moreover, D&F has some Galois theory in it. It can be a little long-winded at times depending on what you’re looking for, but it is there.)

  • $\begingroup$ Thanks a lot ! Do you suggest any among {Marcus, Neurkirch} to read along with this book too ? $\endgroup$ – cdt Apr 5 at 12:16
  • $\begingroup$ Is there some books on Galois Theory written in the same flavor as the book you have mentioned does ? $\endgroup$ – cdt Apr 5 at 12:17
  • $\begingroup$ I haven’t explored Marcus/Nuerkirch much, but I will say that learning math can be fairly fickle and it always helps to have more resources. Pick a book you like, and if you run into problems try looking at the same material in other books to see if their treatment is any more enlightening. Unfortunately, I don’t know of any resources for Galois theory in Baker’s tone off the top of my head, sorry. $\endgroup$ – Santana Afton Apr 5 at 12:32

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