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The other day, in a conversation with colleagues, I realised that the word 'algebra' means different things to us. To me, it brings to mind the study of algebraic structures: vector spaces, groups, rings, fields, algebras, modules... On the other hand, to them it means the things they called algebra at school - mostly rearranging and solving equations. These two ideas feel very different. Is there a good way of explaining how they fit together?

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  • $\begingroup$ An algebraic structure is a context in which one can solve equations? $\endgroup$ – Marius Kempe Jul 4 '15 at 14:38
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    $\begingroup$ There is some common confusion about this. At Ohio State, the abstract algebra course sequence for math majors was called "Algebra I, Algebra II, Algebra III" (back in the days of three terms per year...). But some students would say: I already took Algebra I and II in high school. So they assigned better names to the courses when they re-organized the curriculum into semesters. $\endgroup$ – Gerald Edgar Jul 4 '15 at 15:20
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    $\begingroup$ I think you raise a good point. Many students think that algebra is so named because letters are used instead of numbers. $\endgroup$ – Karl Jul 4 '15 at 17:31
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    $\begingroup$ @Karl: In Dutch, I've noticed that secondary school algebra is often referred to as rekenen met letters (lit. arithmetic/calculating/reckoning with letters). $\endgroup$ – J W Jul 4 '15 at 20:03
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    $\begingroup$ Regarding terminology, Wikipedia says: The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. $\endgroup$ – Joel Reyes Noche Jul 5 '15 at 0:50
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Elementary algebra is the study of a few specific rings: integers, rational numbers, polynomials with rational coefficients, and rational functions. All the rules for "rearranging" equations are applications of the ring or field axioms. Solving equations can be seen in the framework of algebraic or analytic geometry as studying the solution sets in some affine space. So, when presented in a coherent, logically precise way, elementary algebra is the first step toward abstract algebra.

A major conceptual shift from elementary algebra to abstract algebra is in what the basic objects of the theory are. In elementary algebra, the basic objects are individual numbers. In abstract algebra, each whole structure is thought of as a basic object of study. (E.g., in ring theory, we're interested in how various "number systems" relate to and differ from each other, not just in elements of a single number system.)

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The main reasons both of these are called the same thing are historical. Group theory was more or less invented by Galois to study when one could solve polynomial equations by radicals, and ring theory was more or less invented by Hilbert to solve the main problems of invariant theory, which aimed to find ways to discover when two systems of equations were "the same" (meaning equivalent after an appropriate change of coordinates). It took Noether and van der Waarden to put these on a completely abstract footing, but the original motivation was to indirectly answer specific questions about solving equations.

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  • $\begingroup$ I think that a large part of ring theory was also motivated by Fermat's Theorem. $\endgroup$ – GPerez Jul 5 '15 at 18:33
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    $\begingroup$ That is true, but I'm not sure if the move towards framing their definitions abstractly happened independently in number theory or if number theory followed along after Hilbert, Noether, and van der Waarden. As far as I can tell, Kummer or even Dedekind didn't think of themselves as studying rings; they thought they were studying collections of algebraic integers. $\endgroup$ – Alexander Woo Jul 7 '15 at 20:10
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A nice time to discuss this with students is when you talk about matrix operations. Matrices, of course, form a non-commutative ring. You can write out the list of ring axioms: $A+B=B+A$, $A+(B+C)=(A+B)+C$, $A(BC) = (AB)C$, etcetera... and have them all be true statements about matrices, which pointing out that, in general, we don't have $AB=BA$.

Then you can point out that the true axioms imply many of the identities they are used to. For example, $(A+\mathrm{Id})^2 = A (A + \mathrm{Id}) + \mathrm{Id} (A+\mathrm{Id}) = A^2 + A + A + \mathrm{Id} = A^2+2A + \mathrm{Id}$ just like they are used to. But not always! $(A+B)(A-B) = A^2 +BA - AB - B^2$, which is usually not equal to $A^2 - B^2$. All of this just feels like a slightly funky version of high school algebra.

After a bit of time practicing this sort of thing, you can say "if you like thinking about what sort of different laws mathematical operations might obey in different contexts, and what the consequences of those laws should be, this is the subject of what modern mathematicians call Algebra. The introductory course at this university is Math XXX."

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