Studying Mathematics Thread

[GrandizerII]

How far into the abstract algebra did you get? I always really enjoyed Galois theory. That there is a *proof* that 5th degree polynomials can't be solved via radicals is just *fun*. But you need to do quotient rings and some field theory first.

Thanks for your reply, Polymath.  Galois Theory sounds exciting; however, I'm eager to take a break from abstract algebra and explore another topic.  

That said, I've studied the following topics: groups; fundamental theorems of groups; cyclic groups; subgroups; direct products; functions; symmetric groups; equivalence relations and cosets (I really enjoyed this section); counting the elements of a finite group; normal subgroups and quotient groups (this section was neat); homomorphisms (these are cool); homomorphisms and normal subgroups (I enjoyed this section; it covers the isomorphism theorems, which are very neat IMO); Rings (I enjoyed this section).  I'm currently studying subrings, ideals, and quotient rings.  Once I'm finished with this section, I'm either going to pack it in and conclude my self-study or cover the section on ring homomorphisms and then call it quits.

Or maybe keep going? There's still physics and chemistry to cover (for example).

[polymath257]

How far into the abstract algebra did you get? I always really enjoyed Galois theory. That there is a *proof* that 5th degree polynomials can't be solved via radicals is just *fun*. But you need to do quotient rings and some field theory first.

Thanks for your reply, Polymath.  Galois Theory sounds exciting; however, I'm eager to take a break from abstract algebra and explore another topic.  

That said, I've studied the following topics: groups; fundamental theorems of groups; cyclic groups; subgroups; direct products; functions; symmetric groups; equivalence relations and cosets (I really enjoyed this section); counting the elements of a finite group; normal subgroups and quotient groups (this section was neat); homomorphisms (these are cool); homomorphisms and normal subgroups (I enjoyed this section; it covers the isomorphism theorems, which are very neat IMO); Rings (I enjoyed this section).  I'm currently studying subrings, ideals, and quotient rings.  Once I'm finished with this section, I'm either going to pack it in and conclude my self-study or cover the section on ring homomorphisms and then call it quits.

You might try some basic algebraic topology: the fundamental group is readily accessible and leads to lots of interesting ideas.

[robvalue]

I've heard about that proof for quintics and higher. I've not seen it, but I'm blown away that it's possible to prove things like that.

Prove, not "provide evidence for", like wanky science! Who needs it, right guys?

[Kernel Sohcahtoa]

Thanks for your reply, Polymath.  Galois Theory sounds exciting; however, I'm eager to take a break from abstract algebra and explore another topic.  

That said, I've studied the following topics: groups; fundamental theorems of groups; cyclic groups; subgroups; direct products; functions; symmetric groups; equivalence relations and cosets (I really enjoyed this section); counting the elements of a finite group; normal subgroups and quotient groups (this section was neat); homomorphisms (these are cool); homomorphisms and normal subgroups (I enjoyed this section; it covers the isomorphism theorems, which are very neat IMO); Rings (I enjoyed this section).  I'm currently studying subrings, ideals, and quotient rings.  Once I'm finished with this section, I'm either going to pack it in and conclude my self-study or cover the section on ring homomorphisms and then call it quits.

You might try some basic algebraic topology: the fundamental group is readily accessible and leads to lots of interesting ideas.

Polymath, I want to thank you for mentioning Galois Theory and for mentioning the proof about quintics: it seems very interesting that for n is greater than or equal to 5, there is no general formula for finding the roots of nth degree polynomials in terms of radicals.  Getting underneath ideas like these, understanding/enjoying them, and gaining an appreciation for them, are the reasons why I chose to self-study mathematics as a hobby.

I will complete the chapter on ring homomorphisms.  Completing this section, along with the others that I've completed, will be  equivalent to a course in Abstract Algebra I.  Afterwards, I plan on studying the following topics:  polynomials; polynomial rings and fields; Unique Factorization Domains; Extensions of Fields; Normal and Separable Extensions; Galois Theory; Solvability.  Based on my understanding, this second stretch of material will be the equivalent of a course in Abstract Algebra II.    

P.S. You've motivated me to continue my studies of abstract algebra.  Thank you.

[Aliza]

You might try some basic algebraic topology: the fundamental group is readily accessible and leads to lots of interesting ideas.

Polymath, I want to thank you for mentioning Galois Theory and for mentioning the proof about quintics: it seems very interesting that for n is greater than or equal to 5, there is no general formula for finding the roots of nth degree polynomials in terms of radicals.  Getting underneath ideas like these, understanding/enjoying them, and gaining an appreciation for them, are the reasons why I chose to self-study mathematics as a hobby.

I will complete the chapter on ring homomorphisms.  Completing this section, along with the others that I've completed, will be  equivalent to a course in Abstract Algebra I.  Afterwards, I plan on studying the following topics:  polynomials; polynomial rings and fields; Unique Factorization Domains; Extensions of Fields; Normal and Separable Extensions; Galois Theory; Solvability.  Based on my understanding, this second stretch of material will be the equivalent of a course in Abstract Algebra II.    

P.S. You've motivated me to continue my studies of abstract algebra.  Thank you.

We can study along side each other, but I suspect only one of us is going to be bubbling over with glee and enthusiasm. :p

[Kernel Sohcahtoa]

Aliza, out of curiosity, are you getting a degree in mathematics?

[Aliza]

Aliza, out of curiosity, are you getting a degree in mathematics?

Mathematics is more of a means to an end for me.

[polymath257]

You might try some basic algebraic topology: the fundamental group is readily accessible and leads to lots of interesting ideas.

Polymath, I want to thank you for mentioning Galois Theory and for mentioning the proof about quintics: it seems very interesting that for n is greater than or equal to 5, there is no general formula for finding the roots of nth degree polynomials in terms of radicals.  Getting underneath ideas like these, understanding/enjoying them, and gaining an appreciation for them, are the reasons why I chose to self-study mathematics as a hobby.

I will complete the chapter on ring homomorphisms.  Completing this section, along with the others that I've completed, will be  equivalent to a course in Abstract Algebra I.  Afterwards, I plan on studying the following topics:  polynomials; polynomial rings and fields; Unique Factorization Domains; Extensions of Fields; Normal and Separable Extensions; Galois Theory; Solvability.  Based on my understanding, this second stretch of material will be the equivalent of a course in Abstract Algebra II.    

P.S. You've motivated me to continue my studies of abstract algebra.  Thank you.

Galois theory is actually why group theory got going. There is some wonderful structure here relating roots of polynomials, isomorphisms of fields into themselves, and the structure of groups.

In any case, the progression you give looks good.

if you have any questions, feel free to ask! BTW, what book(s) are you using?

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I've heard about that proof for quintics and higher. I've not seen it, but I'm blown away that it's possible to prove things like that.

Prove, not "provide evidence for", like wanky science! Who needs it, right guys?

Well, at least part of the issue is defining what needs to be proved. Making the question of 'solving polynomials' well-defined as opposed to something vaguely felt is non-trivial.

And, of course, that is before you can even attempt the proof. Finding out how the structure of particular groups (alternating groups) relates to solvability of polynomials by radicals is just way cool.

[Kernel Sohcahtoa]

Polymath, I want to thank you for mentioning Galois Theory and for mentioning the proof about quintics: it seems very interesting that for n is greater than or equal to 5, there is no general formula for finding the roots of nth degree polynomials in terms of radicals.  Getting underneath ideas like these, understanding/enjoying them, and gaining an appreciation for them, are the reasons why I chose to self-study mathematics as a hobby.

I will complete the chapter on ring homomorphisms.  Completing this section, along with the others that I've completed, will be  equivalent to a course in Abstract Algebra I.  Afterwards, I plan on studying the following topics:  polynomials; polynomial rings and fields; Unique Factorization Domains; Extensions of Fields; Normal and Separable Extensions; Galois Theory; Solvability.  Based on my understanding, this second stretch of material will be the equivalent of a course in Abstract Algebra II.    

P.S. You've motivated me to continue my studies of abstract algebra.  Thank you.

Galois theory is actually why group theory got going. There is some wonderful structure here relating roots of polynomials, isomorphisms of fields into themselves, and the structure of groups.

In any case, the progression you give looks good.

if you have any questions, feel free to ask! BTW, what book(s) are you using?

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.

[polymath257]

Galois theory is actually why group theory got going. There is some wonderful structure here relating roots of polynomials, isomorphisms of fields into themselves, and the structure of groups.

In any case, the progression you give looks good.

if you have any questions, feel free to ask! BTW, what book(s) are you using?

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.