Studying Mathematics Thread

[Kernel Sohcahtoa]

I didn't realize that Galois theory got group theory going.  I really enjoyed group theory,  so now I have  more motivation to get underneath Galois theory.

I'm primarily using Abstract Algebra by Saracino.  For whatever reason, his method of instruction and his style of proof-writing work well for me; I'm grateful that he has written his book, as it has made abstract algebra pretty accessible to me.  On occasion, I'll open up A Book of Abstract Algebra by Pinter or A First Course in Abstract Algebra by Fraleigh if I find that a particular concept is a little vague in Saracino. Also, if I have any questions and I'm able to express them in a precise and coherent manner (IMO, this is an art-form in itself), then I'll post them here.  Thanks.

Looking at the Pinter book, I had forgotten to mention the questions about ruler and compass constructions, which are related in some interesting ways to Field and Galois theory.

So, the question of which regular polygons can be constructed by ruler and compass has been completely answered (well, except for some number theory issues). The fact that trisecting a general angle and doubling a cube cannot be done via ruler and compass also follows from this directly.

Just scanning the table of contents suggests Pinter's book will cover this.

Definitely feel free to ask questions either publicly or via PM. I'm always willing to help with math.

Thank you, polymath.  Saracino's book also has a section on ruler and compass constructions, which is right in between unique factorization domains and normal and separable extensions.  I'll be sure to add that to my study list, as it sounds cool based on your description.

(1) What attributes do you like about one particular book over another? (2) Do you favor a more narrative description where the author writes like they're speaking to you, or do you like step-by-step solutions with notes on the side telling you what's going on? Or is it another attribute entirely, like examples being more like just general equations to solve vs. examples having a practical, real-world application that you can relate to?

(1) In regards to higher level mathematics (proof-based), I like a book where I am challenged to fill in the missing details of the author's proof: IMO, filling in the details and really getting underneath the material, allows me to gain an appreciation for the subject matter, which enhances the learning process for me.  For example, in Saracino's Abstract Algebra, he writes plenty of proofs that are condensed, and as a result, he may not mention that he negated a particular definition in his proof or that his chosen proof technique is actually the contra-positive of the theorem or of a definition.  Thus, in these cases, the reader would have to carefully analyze each sentence in order to identify key mathematical concepts and proof techniques that would  clarify the author's conclusions, which can seem to come out of nowhere, especially  if one has not taken the time to really get underneath the proof. 

(2) I do like books that sound like the author is talking to you; I'm not talented at math, so I need a book that is accessible and explains the material while still allowing plenty of opportunities for the reader to challenge himself or herself  .  I like Saracino's instructive approach, because once he proves a key theorem in a chapter, he will usually connect the generality of the theorem to specific examples. For example, if Saracino proves a particular theorem about groups, then he may talk about the set of integers, the set of real numbers, the integers modulo n, the set of all 2X2 invertible matrices, etc., and how they have all the properties of the theorem.  Thus, his approach works for me.  


That said, how about you, Aliza? What type of math books do you like?  You mentioned earlier that math is more of a means to an end for you. Would you be willing to elaborate? Thanks for your response.