# Where are the other “algebra” tags besides “abstract” and “precalculus”?

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?

• Could you give some examples? – Noah Snyder Oct 15 '12 at 22:17
• @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. – Gerry Myerson Oct 15 '12 at 23:06
• (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] – Michael Hardy Oct 16 '12 at 2:27
• .... (6) Does Euclid's algorithm for GCDs work with polynomials in three variables? – Michael Hardy Oct 16 '12 at 2:32
• ...... (7) How does one prove?; and (8) How does one use, the Buckingham pi theorem? – Michael Hardy Oct 16 '12 at 2:35
• .....(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 $p_n\,'(0)=c_n$? – Michael Hardy Oct 16 '12 at 2:38
• ....(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 $\le \deg p(x)$? – Michael Hardy Oct 16 '12 at 2:41
• ....(10) Why is there a dot-product only in dimensions $3$ and $7$? – Michael Hardy Oct 16 '12 at 2:46
• .....(11) What's the difference between an ordered pair of vectors and a tensor product of two vectors? – Michael Hardy Oct 16 '12 at 2:48
• ....(12) How do you completely factor $x^n-1$ into polynomials with integer coefficients? – Michael Hardy Oct 16 '12 at 2:52
• 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". – Gerry Myerson Oct 16 '12 at 5:44
• 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". – Willie Wong Oct 16 '12 at 7:06
• 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. – J. M. isn't a mathematician Oct 16 '12 at 12:06
• @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 $f_n'(x)=nf_{n-1}(x)$. – Michael Hardy Oct 16 '12 at 18:10
• 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". – Michael Hardy Oct 16 '12 at 18:12

Other "algebra" tags include (currently 4940 questions with this tag), (2555), (2200), (1363), (1164), (998), (707), (634), (590), (579), (466), (389), (361), (254), (241), (191), (187), (175), (172), (171), (167), (165), and many, many more.

• That's a good argument for removing the abstract-algebra tag. – zyx Oct 17 '12 at 13:25
• 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. – Michael Hardy Oct 17 '12 at 18:46
• 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. – Michael Hardy Oct 17 '12 at 18:47
• Forgot to mention universal-algebra. – Incnis Mrsi Dec 13 '14 at 13:17

On math.SE, “abstract algebra” is commonly used as a substitute for one of three things: , , and . Possibly many question also fall to this sinkhole. More specific tags as (pertain to algebras) and (pertains to ring theory) should be used as well, but a question tagged with only 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 , although questions tagged so at math.SE demonstrate significant pollution with not-really-universal things mentioned above.

• 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. – MJD Dec 13 '14 at 16:44