What Theoretical Algebra Book Covers Advanced Topics Beyond High School Level?

[Kalvino]

What is a good book on algebra that is highly theoretical and covers functions, the binomial theorem, sequences/series (basically all algebra topics in college) and if possible, elementary number theory?

Most books on high school-college algebra today don't cover theory at all and when they do claim to do so, they really don't. Books today are fluffed up with the "How?" rather than the "Why?" and I find this very infuriating. That's why I am looking for a theoretical algebra book, IF it even exists.

NOTE: Gelfand's Algebra is too elementary and I have already read it.

 

[axmls]

What you're describing is an introductory abstract algebra textbook (typically Junior in college-level). I've heard good things about
Pinter's "A book of abstract algebra."

It's a Dover book, so it's available for 10-20$.

However, it's good to have some mathematical maturity before diving into abstract algebra.

Note: most of the topics one finds in high school algebra are taught on a more theoretical basis in analysis textbooks. To a lesser extent, in calculus textbooks (things like slope, etc.)

 

[MidgetDwarf]

What you're describing is an introductory abstract algebra textbook (typically Junior in college-level). I've heard good things about
Pinter's "A book of abstract algebra."

It's a Dover book, so it's available for 10-20$.

However, it's good to have some mathematical maturity before diving into abstract algebra.

Note: most of the topics one finds in high school algebra are taught on a more theoretical basis in analysis textbooks. To a lesser extent, in calculus textbooks (things like slope, etc.)

you can get a hard cover of Pinter's book for 5 dollars shipped.

 

[mathwonk]

I like Euler's Elements of Algebra. but you should look at it in a library or somewhere before investing money. You might not like it. It does have a lot of elementary stuff but also gets eventually quite advanced, and there is a more advanced addendum by Lagrange. It is also available free online.

 

[Kalvino]

I like Euler's Elements of Algebra. but you should look at it in a library or somewhere before investing money. You might not like it. It does have a lot of elementary stuff but also gets eventually quite advanced, and there is a more advanced addendum by Lagrange. It is also available free online.

I am glad you answered, mathwonk. Gladder than I have ever been before, since it was originally your suggestion on here that convinced me to buy Euler's Elements of Algebra a week ago. However there is one problem, it does not introduce the concept of a f(x)-type function. I was looking for a similar book with the same theoretical vibe and brilliant exposition that delves a little further from Euler's book. After reading Elements by Euler, what do you think I should read? (I have read Euclid's Elements already) - It would be quite nice if there happened to be a book by Euler that discussed functions, sequence/series, binomial theorem, matrices/determinants since he is absolutely brilliant. But I am not sure if there is any, do you have something similar with the topics I mentioned in mind? If yes, please let me know.

 

[mathwonk]

well functions are mentioned in a footnote on page 465 by Lagrange in the addendum to that book, but not the notation f(x). Euler has another book, Introduction to analysis of the infinite, sor of his precalculus book, which studies functions more generally but I again do not see the notation f(x) there. Of course that does not bother me as the notation is only incidental to the subjct, but I am puzzled since I always heard it was Euler who introduced that notation. On th other hand Euler was mostly concerned with functions that have explcit formulas, or at least implicit ones, so did not need to use an abstract functional notation like f(x). He just gave the definition of his functions.

 

[mathwonk]

http://jeff560.tripod.com/functions.html

 

[Vardaan Bhat]

It's not the most theoretical book, but it is amazing: Art of problem Solving's intermediate algebra

 

[MidgetDwarf]

It's not the most theoretical book, but it is amazing: Art of problem Solving's intermediate algebra

that is to basic for the thread starter's reasons.

 

[gleem]

What is a good book on algebra that is highly theoretical and covers functions, the binomial theorem, sequences/series (basically all algebra topics in college) and if possible, elementary number theory?

This content sounds more like analysis to me the study of continuously varying quantities or numbers like functions, series.and sequences. For algebra ( of which there are many ) generally is the study of mathematical systems where a mathematical systems consists of undefined terms, elements and operations, axioms and postulates that establish relationships between the two, and definitions and theorems.

 

[mathwonk]

you might like "Principles of Mathematics" by Allendoerfer and Oakley, not an algebra book exactly but full of lots of useful stuff up to calculus. See if you can find one in a library. In fact her is one for under $6.

http://www.abebooks.com/servlet/SearchResults?an=allendoerfer,+oakley&sts=t