What is "Abstract Algebra"? How is it different than High School Algebra?

Answer 1

You can think of high school algebra as learning the language of mathematics. Abstract algebra actually uses that language to study symmetries (which is group theory), the nature of factorization (ring theory) and of the roots of polynomials (field theory). So while high school algebra is more devoted to techniques for solving specific types of problems, abstract algebra is devoted to proving the underlying theory behind algebra.

Answer 2

You may want to look at the book Abstract Algebra and Solution by Radicals by John and Margaret Maxfield. It is a good transition book. It shows how some of the ideas developed in High School algebra extend naturally to Abstract Algebra.

Answer 3

Abstract Algebra is also known as Modern Algebra and in the 1970s, was being introduced to kids in elementary school as the "new math". However, there were many criticisms of trying to teach this to grade-schoolers when the "basics" of mathematics hadn't been learned yet and the fact that abstract algebra was a subject for college material.

Abstract algebra is not the algebra that is often taught in high school and is very elementary in its substutitution of letters for numbers to generalize equations and ideas such as x + y = z. Substitution of letters for numbers is a form of universalization of ideas in mathematics and high-school algebra introduces this concept. High-shool algebra will often introduce ideas for this concept of universalization by practical examples such as "word stories" (how much older would Sally be than John if she was twice as old as John three years ago, how tall is this structure if the shadow is at this angle, etc.) In essence, the "algebra" classes that are taught in high school as part of "geometry, trigonometry, etc." are watered-down versions and non-encompassing of things included in abstract algebra.

Abstract algebra is often denoted as just "algebra" but, because it can become confused with the watered-down versions taught in high school and often, in elementary school, "abstract algebra" or "modern algebra" is used for more precise clarification. It is entirely theoretical and heavily proofed (there are many proofs).

Abstract Algebra encompasses things known as group theory, ring theory, number theory (another field of mathematics) and, essentially, attempts to touch on all aspect of things mathematical. For instance, group theory and ring structure attempt to encompass and classify all number bases and how they behave. For instance, we in the Western world use the base 10 system, but group theory attempts to put all bases (base 2 is a binary system used much in computers and has two symbols, the hexadecimal has 16 symbols and is used in identifying scientific systems, and base 10 has 10 symbols - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 - and is the easiest for many in the West to comprehend numbers) in the study of numbers and how those numbering systems behave. As an aside, many American Indian groups used different base systems for numbering other than base 10, such as the Incas in South America.

One of the earliest proponents and pioneers of abstract algebraic studies was the mathematician Garrett Birkhoff, how wrote the first major works in America on the subject. You can find more out about him at http://en.wikipedia.org/wiki/Garrett_Birkhoff

Answer 4

it is very different
it's about groups, rings, fields, etc which are generalized algebraical structures
for example the set of rational numbers with mutiplication is a group, with addition is also a group
also you learn about matrices and permuatation