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When I taught courses on algebra giving a first exposition to Galois theory I usually included some discussion of classical results showing the impossibility of constructing certain points with ruler and compass, such a the cube root of $2$ (Delian problem) and the trisection of a (generic) angle.

However, sometimes I think maybe this is not that good an allocation of time. It took me quite some time to recall and/or to introduce in precise terms what it means to construct something, which feels a bit like an isolated subject in such a course. Likewise, the results, while historically important and interesting, do not seem of much use later on. Since as said I did include the subject, needless to say, I can also see some merit in it. Yet, I never quite managed to make up my mind.

Thus, I would like further input on this subject:

What are reasons for or against teaching the impossibility of certain geometric constructions in a course on algebra that covers the basics of Galois theory?

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    $\begingroup$ These are proved in Cox's Galois Theory as Examples 10.1.9, 10.1.10, and (squaring the circle) 10.1.11. The lead-up only takes about 6 pages, after which they are all captured in one fell-swoop (though they refer back to earlier examples/theorems; the one accept-without-proof, for squaring the circle, is that $\pi$ is transcendental). Looking back over this material, and depending on class size / interest, I wonder whether you could have a day in which students presented (in small groups) on these three examples. Practice "filling in the details" might be a reason to spend time on them. $\endgroup$
    – Benjamin Dickman
    Oct 18, 2015 at 23:29
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    $\begingroup$ When I took the required course covering Galois theory, it seemed to me like the most useless and silly thing I had ever studied. The classical results you're talking about were the only applications we ever saw that gave me any hint of why anyone would care about the subject -- and even they seemed to me like extremely weak motivation. If you didn't use these topics as motivation, what would you tell your students was the reason they should care about Galois theory? $\endgroup$
    – user507
    Oct 19, 2015 at 1:53
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    $\begingroup$ @BenCrowell there'd remain what is arguably the original motivation namely the (un)solvability of polynomial equations by radicals. But I agree that the classical results are interesting. $\endgroup$
    – quid ♦
    Oct 19, 2015 at 22:42
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    $\begingroup$ Will you choose whether the question is about "a course on Galois theory" (as at the end of the post), or "a course on algebra" (as in the title)? For the first, I'd include these classical topics; for the second, I'd replace Galois theory with other algebraic topics entirely. So my answer depends on which question you're asking. $\endgroup$
    – user173
    Oct 20, 2015 at 14:56
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    $\begingroup$ @MattF. thanks I agree it was unclear, I edited it a bit It is somewhat in the middle; the context is a course on algebra that includes basics of Galois theory, basically builds up to them. Not to cover Galois theory is a non-option but it could be deemphasized in order to have a time for other things. (The later is basically what I am thinking about and motivated this question.) $\endgroup$
    – quid ♦
    Oct 20, 2015 at 15:08

2 Answers 2

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Let me first say how I have taught this, and then why it was worth doing.

Here is the stripped down version I speak of. The first block of bullet points is one day.

  • Forget about straight edge and compass, no one uses them. Fortunately, I think we still have another few years of students who have still seen the standard five function calculator: $+$, $-$, $\times$, $\div$, $\sqrt{ \ }$. So the question was: using these keys, and the ability to type in integers or recopy numbers that you have computed before, what can you compute? Can you compute $\sqrt[3]{2}$, or $\cos (20^{\circ})$?

  • We'll show that the answer is no! Suppose that we could compute $\sqrt[3]{2}$. Let $\theta_1$, $\theta_2$, ..., $\theta_N$ be the sequence of numbers displayed on our calculator. So each $\theta_i$ is made up of one or two previous $\theta_i$, using the five operations above.

  • Let $K_i$ be the field $\mathbb{Q}(\theta_1, \theta_2, \ldots, \theta_i)$. Then $[K_{i+1}:K_i]=1$ or $2$ for each $i$. So $[K_N:\mathbb{Q}]=2^r$ for some $r$.

  • If $\sqrt[3]{2} \in K_N$, then $\mathbb{Q}(\sqrt[3]{2}) \subseteq K_N$. But then $3 = [\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]$ divides $2^r$, a contradiction.

  • Tell students that the historical version of this problem is using straight-edge and compass, not calculators.

Here ends that day, but later follow up:

  • After we have introduced Galois groups, have them prove the converse: If $\mathrm{Gal}(K/\mathbb{Q})$ is a $2$-group, then elements of $K$ can be constructed using the $5$-function calculator.

  • Work out the construction of the $17$-gon using the above.

  • When we get to solvability, point out that "$2$-group" means "all Jordan-Holder constituents are $\mathbb{Z}/2\mathbb{Z}$", and that we are now looking for an analogous description of fields that involve any radical extension.

The most recent time I taught this course, I did it as an IBL class. Here is the main worksheet corresponding to this material.

So, why do I think this was worth it?

  • I think it is a moderately interesting question, and it can be attacked just using basic field theory tools like degree of an extension and minimal polynomials. It's nice to reach an application before we bring in the Galois groups. If you've never seen it done, the idea of proving that no formula exists can seem like a miracle, and the quintic case is hard enough that I don't think most students internalize the full argument.

  • The story about sequentially computing a sequence of numbers on your calculator makes the tower of field extensions natural.

  • The followup of proving that "Galois group is a $2$-group" implies "tower of square root extensions" is a good preparation for Kummer theory. It also makes the question "what criterion describes a tower of radical extensions" natural.

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    $\begingroup$ This is a creative solution! I remember it from a previous comment of yours. Do you have an opinion on the broader question -- do you think Galois theory merits a place in a one-year undergraduate algebra curriculum, or would you rather emphasize material more relevant to coding, computer graphics, cryptology, physical symmetries? See here for discussion: math.stackexchange.com/questions/449066 $\endgroup$
    – user173
    Oct 20, 2015 at 20:37
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    $\begingroup$ The calculator motivation is a good idea, thanks for sharing it. Re what @MattF. said: matheducators.stackexchange.com/questions/2612/… is also related but focused on a specific context. Maybe there is room for another question along these lines. $\endgroup$
    – quid ♦
    Oct 20, 2015 at 20:48
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    $\begingroup$ On the one hand, I love this idea. On the other hand, none of my students has a calculator with a square-root button. Instead, they have calculators with something like a square-root-rounded-off-to-thirteen-digits button. I'd expect the brighter ones to realize that your analysis is irrelevant to the question of whether their calculators can accurately find the first thirteen digits of the cube root of 2. $\endgroup$
    – WillO
    Sep 27, 2016 at 16:38
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Perhaps rather than spend time establishing that trisecting an angle is impossible via Euclidean (ruler-compass) constructions, you could instead (a) Make that claim without proof, and (b) Mention that different axioms do permit angle trisection:

    Fig.19.1
          (Figure from Geometric Folding Algorithms: Linkages, Origami, Polyhedra, p.286.)
This would (temporarily) turn the topic away from Galois theory and toward the history and philosophy of mathematics, which could be an illuminating interlude. You could even have the students trisect an angle in-class via Abe's construction! :-) ~15 minutes.

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    $\begingroup$ That's an interesting idea. It also makes me wonder what (if any) the characterization of the points constructible in this way would be. (But this is another question.) $\endgroup$
    – quid ♦
    Oct 18, 2015 at 20:56
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    $\begingroup$ @quid: Alperin, Roger C. "A mathematical theory of origami constructions and numbers." New York J. Math 6.119 (2000): 133. (PDF download.): "In this article we give a simplified set of axioms for mathematical origami and numbers. The axioms are hierarchically structured so that the addition of each axiom, allowing new geometrical complications, is mirrored in the field theory of the possible constructible numbers. The fields of Thalian, Pythagorean, Euclidean and Origami numbers are thus obtained using this set of axioms." $\endgroup$
    – Joseph O'Rourke
    Oct 18, 2015 at 21:00
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    $\begingroup$ Thanks a lot for this reference. This looks very interesting for my purpose. $\endgroup$
    – quid ♦
    Oct 18, 2015 at 21:03
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    $\begingroup$ @quid I'm not sure what text you are using, but you might check Cox's Galois Theory. Chapter 10 is entitled "Geometric Constructions" and includes an optional section, 10.3, called "Origami." (The first reference provided in that section is the R.C. Alperin paper suggested by JO'R above!) $\endgroup$
    – Benjamin Dickman
    Oct 18, 2015 at 23:20
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    $\begingroup$ @quid: See Some Remarks on Conic Constructible Numbers on pp. 6-7 of my manuscript A Detailed and Elementary Solution to $x^{17}=1$. Briefly, these are numbers that can be obtained by using finite sequences of the four arithmetic operations along with the operation of "solving cubic equations" starting with the rationals, in the same way that the constructible numbers can be obtained by using finite sequences of the four arithmetic operations along with the operation of "solving quadratic equations" starting with the rationals. $\endgroup$
    – Dave L Renfro
    Oct 23, 2015 at 17:59

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