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When studying mathematics at university in the USA there is often a course called "Abtract Algebra". I get that the course for many is the first time one experiences very abstract high level mathematics.

What does the "Abstract" here mean?

Are there other types of Algebra? I get that there is "algebra" in 8th grade. But the two don't seem very related.

Mike
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John Doe
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    I actually prefer "Modern Algebra" instead of "Abstract Algebra". – Fei Cao Mar 08 '22 at 20:42
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    This Wikipedia section lists several areas with "algebra" in their name. The two that you mention in your question are referred to as abstract algebra / modern algebra and elementary algebra, although maybe someone else can comment on the etymology of the "abstract" adjective in this context. – angryavian Mar 08 '22 at 20:45
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    The algebra in primary school is still "abstract algebra"-- after all, questions of where roots of polynomials are are addressed first in primary school, then again in field theory. I am not aware of a historical reason, but I interpret the "abstract" part to denote that all results in primary school are concrete and explicit, but this is not the case in higher mathematics. – While I Am Mar 08 '22 at 20:47
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    Secondary school algebra is over the reals. You then "abstract" those properties to other sets and operations. That is abstract algebra. – Randall Mar 08 '22 at 21:00
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    @William: In the U.S. the term "primary school" often refers to grades K-2 or K-3 (ages 5-8; first few grades of elementary school), many years before a student will see any kind of algebra (unless the student is extremely gifted/advanced). – Dave L. Renfro Mar 08 '22 at 21:01
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    Here "abstract" means we've abstracted out common properties of concrete algebraic structures into axioms that can be used to specify classes of algebraic structures that generalize these concrete structures, e.g. the ring axioms generalize the algebraic properties of many concrete algebraic systems encountered in elementary (pre-university) algebra (including number rings, and rings of polynomials, power series, matrices, etc). This is an instance of the "axiomatic method" (search on that to learn more). See also the Remark in this answer. – Bill Dubuque Mar 08 '22 at 22:27
  • @Dave L. Renfro Ah, that's right. I meant secondary school. – While I Am Mar 09 '22 at 15:02

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