If the "algebra" tag is declared heretical, as suggested here, then what should one do with algebra questions for which neither the "abstract algebra" tag nor a "precalculus" tag is appropriate?
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4$\begingroup$ Could you give some examples? $\endgroup$– Noah SnyderCommented Oct 15, 2012 at 22:17
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3$\begingroup$ @Noah, there has recently been a run on questions about discriminants of polynomials. I'm not comfortable with either the abstract-algebra tag or the algebra-precalculus tag for these. But maybe the "polynomials" tag would do. $\endgroup$– Gerry MyersonCommented Oct 15, 2012 at 23:06
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$\begingroup$ (1) discriminants of polynomials; (2) How does one show that the product of two sums of squares of integers is a sum of squares in at least two different ways?; (3) How does one prove the Cayley--Hamilton theorem for real matrices?; (4) How does one use the singular-value decomposition in signal processing?; (5) Suppose $X$ is a $1000\times 2$ matrix of rank $2$ and you've shown that $X\hat\beta=X(X^T X)^{-1}X^T Y$. How do you find a "left inverse" of the non-square matrix $X$ so that you can justify the left-cancellation that tells you that $\hat\beta=(X^T X)^{-1} X^T Y$? [to be continued] $\endgroup$– Michael HardyCommented Oct 16, 2012 at 2:27
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$\begingroup$ .... (6) Does Euclid's algorithm for GCDs work with polynomials in three variables? $\endgroup$– Michael HardyCommented Oct 16, 2012 at 2:32
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$\begingroup$ ...... (7) How does one prove?; and (8) How does one use, the Buckingham pi theorem? $\endgroup$– Michael HardyCommented Oct 16, 2012 at 2:35
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$\begingroup$ .....(8) Is it true that if $c_1,c_2,c_3,\ldots$ is any sequence of scalars, there is exactly one sequence of polynomials $p_n(x)$, $n=1,2,3,\ldots$ such that for all $n$, $\deg p_n(x)=n$ and $p_n(x+y)=\sum_{k=0}^n \binom{n}{k} p_k(x)p_{n-k}(y)$ and $p_n\,'(0)=c_n$? $\endgroup$– Michael HardyCommented Oct 16, 2012 at 2:38
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$\begingroup$ ....(9) Why does every shift-equivariant linear operator on the vector space of polynomials in $x$ map every polynomial $p(x)$ to a polynomials whose degree is $\le \deg p(x)$? $\endgroup$– Michael HardyCommented Oct 16, 2012 at 2:41
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$\begingroup$ ....(10) Why is there a dot-product only in dimensions $3$ and $7$? $\endgroup$– Michael HardyCommented Oct 16, 2012 at 2:46
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$\begingroup$ .....(11) What's the difference between an ordered pair of vectors and a tensor product of two vectors? $\endgroup$– Michael HardyCommented Oct 16, 2012 at 2:48
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$\begingroup$ ....(12) How do you completely factor $x^n-1$ into polynomials with integer coefficients? $\endgroup$– Michael HardyCommented Oct 16, 2012 at 2:52
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3$\begingroup$ Michael, I'd say that (3), (5), possibly (9) could go linear-algebra; (1), (12), (17), (19) and maybe some others could be tagged "polynomials"; (2), (15) number-theory; (13) is logic; perhaps there's an "identities" tag that can be used, perhaps a commutative-algebra tag; it takes some imagination, but I suspect that for each of these there's a more informative tag than "algebra". $\endgroup$– Gerry MyersonCommented Oct 16, 2012 at 5:44
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1$\begingroup$ For (7) and [the first, as there are two] (8) we almost certainly should create a "dimensional-analysis" tag, and the "physics" tag would also be appropriate. (4) is obviously "linear-algebra" + "signal-processing", perhaps with a dose of "numerical-methods". (6) is perfectly fine for "abstract-algebra". (20) is good for some statistics tag + "error-propagation". (11) is good for "multilinear-algebra". $\endgroup$– Willie Wong ModCommented Oct 16, 2012 at 7:06
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1$\begingroup$ For 14: that's why there's orthogonal-polynomials. It could also be tagged special-functions. 20 will fall under statistics and regression. 17 could also have math-history in addition to polynomials. $\endgroup$– J. M. ain't a mathematicianCommented Oct 16, 2012 at 12:06
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1$\begingroup$ @J.M. : I'm not sure orthogonality is of the essence in #14. The commutativity in question applies to any two Appell sequences, which are polynomial sequences satifying $f_n'(x)=nf_{n-1}(x)$. $\endgroup$– Michael HardyCommented Oct 16, 2012 at 18:10
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1$\begingroup$ I think at least some of my examples escape other tags. The stuff about discriminants was already pointed out by Gerry Myerson: It's algebra, but it's not "abstract" (i.e. about which algebraic structures satisfy which axioms, etc.) nor "precalculus". $\endgroup$– Michael HardyCommented Oct 16, 2012 at 18:12
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2 Answers
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Other "algebra" tags include linear-algebra (currently 4940 questions with this tag), group-theory (2555), matrices (2200), commutative-algebra (1363), polynomials (1164), ring-theory (998), field-theory (707), vector-spaces (634), representation-theory (590), finite-groups (579), modules (466), galois-theory (389), eigenvalues (361), homological-algebra (254), finite-fields (241), roots (191), abelian-groups (187), eigenvectors (175), determinant (172), numerical-linear-algebra (171), relations (167), boolean-algebra (165), and many, many more.
answered Oct 17, 2012 at 5:13
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1$\begingroup$ That's a good argument for removing the abstract-algebra tag. $\endgroup$– zyxCommented Oct 17, 2012 at 13:25
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$\begingroup$ I find this answer inadequate because what I really had in mind was just algebra with real or complex numbers that can be done by people who've never heard of groups, rings, etc., but that is far too advanced to qualify as pre-calculus and that at any rate would be inappropriate in a course preparing students for calculus. Gerry Myerson mentioned some stuff about discriminants, and maybe that's the only good example mentioned here so far. My hastily compiled list of examples was indeed hastily compiled. $\endgroup$ Commented Oct 17, 2012 at 18:46
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1$\begingroup$ Besides, many of these tags you list are topics included within what people call "abstract algebra", so that disqualifies them as examples of appropriate tags for things where "abstract algebra" is inappropriate. $\endgroup$ Commented Oct 17, 2012 at 18:47
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On math.SE, “abstract algebra” is commonly used as a substitute for one of three things: group-theory, ring-theory, and algebras. Possibly many boolean-algebra question also fall to this sinkhole. More specific tags as polynomials (pertain to algebras) and field-theory (pertains to ring theory) should be used as well, but a question tagged with only abstract-algebra is something abnormal.
The whole “abstract algebra” umbrella (as well as its “algebra-precalculus” counterpart) seemingly originates from classification of educational courses in some countries, allegedly USA.
There is also such concept as universal-algebra, although questions tagged so at math.SE demonstrate significant pollution with not-really-universal things mentioned above.
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3$\begingroup$ A web search for "abstract algebra" will produce many textbooks, lecture notes, and course listings (in the USA and elsewhere) that have that title. The fact that your preferred terminology for this material is different does not render the term "abstract algebra" any less legitimate or correct. $\endgroup$– MJDCommented Dec 13, 2014 at 16:44