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As I was reading over Galois' wikipedia page, I noticed that he (Galois) read a book called "Réflexions sur la résolution algébrique des équations" at age 15, which I then read has to do with the Lagrange resolvent of polynomial equations, and from there I somehow got to the fundamental theorem of algebra blah blah blah you know how wikipedia is...

And to make a long story short, I realized that there seems to be a heck of a lot of very complex, interesting-looking algebra (especially on the proof side of things) that I didn't even get close to in Algebra 1 and 2 back before I even knew what a proof was. I'm just wondering whether it's worth it to go back and look over some of the "advanced algebra" the we glossed over in early high school.

I recently finished calc 1 and I'm currently learning some basic set theory/logic/proof-writing stuff (reading How to Prove it) because I started to get interested in proof construction.

I plan to read Spivak's this summer (I'll be a freshman in college next fall), but is it worth it to also go back to stuff the we only learned superficially in 9-11th grade or will I get into that in abstract algebra and whatnot in the years to come?

Sorry about how long this was… the background on my excursions on wikipedia was probably unnecessary.

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  • $\begingroup$ Lagrange's book is short and accessible. It is important historical background material for Galois theory, but nowadays primarily of historical interest. $\endgroup$ – André Nicolas May 30 '14 at 0:43
  • $\begingroup$ See also math.stackexchange.com/a/152824/589. $\endgroup$ – lhf May 30 '14 at 1:16
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What you read in high school is necessary for college level math. Abstract algebra assumes you know the following: set theory, logic, proof techniques, functions and relations, induction, cardinal numbers, and number theory.

Abstract algebra covers groups, rings, fields, modules, vector spaces, and algebras. Hence, as I said above, functions and relations, which is what you learn in high school, is required for abstract algebra.

Most universities do not teach precalculus, assuming that the students are familiar and comfortable with it. It is important to know at least that much, since it'll help you in calculus of a single and of multi variables.

Good luck!

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