Bargain-Basement Mathematics

April 19, 2023

Posted by John Baez

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The fundamental theorem of Galois theory and that fundamental theorem of algebraic geometry called the Nullstellensatz are not trivial, at least not to me. But they both have cheaper versions that really are. So right now I’m pondering the difference between ‘bargain-basement mathematics’ — results that are cheap and easy — and the more glamorous, harder to understand mathematics that often gets taught in school.

Since I’m talking about bargain-basement mathematics, I’ll do it in an elementary style, at least at first — since I want beginners to follow this! I hope experts will look the other way.

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There’s a simple trick that’s fundamental both in Galois theory and algebraic geometry:

1) In Galois theory we have a little field $k$ sitting in a big field $K$ . We want to understand all the fields between them, say $k \subseteq L \subseteq K$ . We study them using the group $Gal(K|k)$ which consists of all symmetries of $K$ that fix every element of $k$ . We note:

Any field $k \subseteq L \subseteq K$ gives a subgroup $G$ of $Gal(K|k)$ , consisting of symmetries that fix every element of $L$ .
Any subgroup $G$ of $Gal(K|k)$ gives a field $k \subseteq L \subseteq K$ , consisting of the elements of $K$ fixed by all the symmetries in $G$ .

These ways of going from $L$ to $G$ and from $G$ to $L$ are not inverses, in general. But they are flip sides of the same coin — and either way, the bigger the field $L$ the smaller the group $G$ . Analyzing this business we get a bunch of Galois theory.

2) In algebraic geometry we study sets where a bunch of polynomials vanish. so we let $k[x_1, \dots, x_n]$ be the polynomials in $n$ variables with coefficients in some field $k$ , and let $k^n$ be $n$ -dimensional space, where points are $n$ -tuples of guys in $k$ . We note:

Any subset $S$ of $k[x_1, \dots, x_n]$ gives a subset $V$ of $k^n$ , consisting of the points where all the polynomials in $S$ vanish.
Any subset $V$ of $k^n$ gives a subset $S$ of $k[x_1, \dots, x_n]$ , consisting of the polynomials that vanish at all the points of $V$ .

These ways of going from $S$ to $V$ and from $V$ to $S$ are not inverses, in general. But they are flip sides of the same coin — and either way, the bigger the set $S$ the smaller the set $V$ . Analyzing this business we get a bunch of basic algebraic geometry.

Notice that we’re using the same trick in each of these two examples. This trick is called a Galois connection. To category theorists, a Galois connection consists of adjoint functors between two partially ordered sets, or ‘posets’, with one of these posets turned upside down. But we don’t even need to understand categories or adjoint functors to get the idea.

Let’s forget about the fact that one poset is turned upside down, since we can always put that fact back in later. Then a Galois connection consists of posets $X$ and $Y$ and order-preserving maps

$L: X \to Y$ $R: Y \to X$

that obey

$L(x) \le y \; iff \; x \le R(y)$

These maps don’t need to be inverses! But we can automatically improve the situation so they are inverses!

To do this ‘improvement’, we simply throw out all elements of $X$ except those in the range of $R$ , and throw out all elements of $Y$ except those in the range of $L$ .

On these new smaller versions of the sets $X$ and $Y$ , the maps $L$ and $R$ become inverses! You can prove this simply by fiddling around with the definition of Galois connection. If you’ve never done it before, it takes some thought. But it’s quite rewarding when you succeed, so please try it.

Doing this ‘improvement’ automatically forces us to discover interesting ideas, so we do it in both Galois theory and algebraic geometry.

For example: in algebraic geometry, a subset $V$ of $k^n$ where some set of polynomials vanishes is called an ‘algebraic set’. And an algebraic set that’s not the union of two smaller ones is called an ‘affine algebraic variety’.

Similarly, a subset $S$ of $k[x_1, \dots, x_n]$ consisting of polynomials that vanish on some subset of $k^n$ is always closed under addition, and under multiplication by anybody in $k[x_1, \dots, x_n]$ . A subset with these properties is called an ‘ideal’. Moreover if any power of some polynomial is in $S$ , that polynomial is in $S$ , so $S$ is called a ‘radical ideal’.

All these concepts are fundamental to algebraic geometry. And we discovered them just by trying to improve a Galois connection!

If we go further with this improvement process, we might eventually rediscover Hilbert’s famous Nullstellensatz. This says that when $k$ is nice enough, our Galois correspondence actually restricts to a bijection between algebraic sets in $k^n$ and radical ideals in $k[x_1, \dots, x_n]$ .

Why do we need $k$ to be ‘nice enough’? And why is the Nullstellensatz nontrivial? Why isn’t it just a spinoff of the basic bargain-basement fact that you can improve any Galois connection to a bijection?

That’s a good puzzle, if you haven’t thought about this stuff. So maybe avert your eyes and think about it.

Here’s the answer.

We defined an algebraic set to be a subset of $k^n$ on which all polynomials in some set $S \subset k[x_1, \dots, x_n]$ vanish. That was smart, if we were aiming to improve our Galois connection to a bijection with a minimum of effort.

But we did not define a radical ideal to be a subset of $k[x_1, \dots, x_n]$ consisting of all polynomials that vanish on some set $V \subset k^n$ . Instead, we just noticed a few random properties that such a subset of $k[x_1, \dots, x_n]$ must have! Then we hastily made those properties into the definition of ‘radical ideal’.

That was a bad move, if our goal was to improve our Galois connection to a bijection with minimum effort. We have to really work to show our Galois connection gives a bijection between algebraic sets and radical ideals. And we can only do it at all when $k$ is sufficiently nice (namely, ‘algebraically closed’.)

In short, if our eyes are firmly focused on the famous Nullstellensatz, which takes some real work to prove, we may miss out on the bargain-basement version sitting right next door. And there’s a similar story at work in Galois theory. The usual treatment gives an impressive result, the fundamental theorem of Galois theory — but it only holds if the field $k$ is sitting in the big one $K$ in a nice way, and it takes some work to prove.

There are definitely advantages to the more expensive results! But I’m thinking that if I ever teach these things, I’ll start out with the bargain-basement versions, and then talk about how to polish them up.

Question: suppose $k$ is any field, not necessarily algebraically closed. What can we say about the subset of guys in $k[x_1, \dots, x_n]$ that vanish on some subset $V \subset k^n$ , apart from the fact that it’s a radical ideal? Can we actually characterize these subsets in the language of ideal theory in any interesting way, or is it just hopeless?

Posted at April 19, 2023 9:32 PM UTC

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Re: Bargain-Basement Mathematics

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I don’t know if this comment belongs in the bargain basement as well, but both of your examples have, on some level, an even more basic commonoality which we’ve discussed before, but I’ll mention for interested Café patrons: both of the examples arise from a relation between two sets.

In the Galois theory example you have the underlying set of $K$ and you have the set $Aut(K)$ of automorphisms of $K$ . The relation $T$ between $K$ and $Aut(K)$ is that $x\in K$ is related to $\sigma \in Aut(K)$ if $x$ is fixed by $\sigma$ : write

$x T \sigma\quad \Leftrightarrow\quad \sigma(x) = x.$

In the algebraic geometry example you have the underlying set of $k^n$ and the underlying set of $k[x_1,\dots , x_n]$ . The relation $T$ between $k^n$ and $k[x_1,\dots , x_n]$ is that $v\in k^n$ is related to $p\in k[x_1,\dots , x_n]$ if $p$ vanishes at $v$ : write

$v T p \quad \Leftrightarrow\quad p(v) = 0.$

Now if we have two sets $G$ and $M$ and a relation $T$ between them then we get a Galois connection

$L\colon subsets(G) \to subsets(M)$ $R\colon subsets(M) \to subsets(G)$

where the set of subsets of a set is given the usual ordering by inclusion.

These are defined as follows. If $V\in subsets(G)$ then $L(V)\in subsets(M)$ consists of the elements of $M$ which are each related to all elements of $V$ :

$m \in L(V) \quad \Longleftrightarrow \quad v T m \quad \forall v\in V.$

The other function $R$ is defined in the same way.

There’s a subtle difference here with what I think you said John, in that the domain here will be, say, subsets of your field $K$ rather than subfields; however, the image of these maps will be structured in the way you said, so when you do the ‘improvement’ and just restrict to the ranges, then you get the same thing.

(If Café patrons want to read more about this sort of thing then they can look at my old posts on Formal Concept Analysis and Classical Dualities and Formal Concept Analysis.)

Posted by: Simon Willerton on April 20, 2023 11:17 AM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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I like your approach, Simon! And that approach quickly leads us to formal concept analysis, which is interesting to an audience who may not be interested in Galois theory or algebraic geometry.

Indeed I considered talking like this:

Any field $k \subseteq L \subseteq K$ gives a subset $G$ of $Gal(K|k)$ , consisting of symmetries that fix every element of $L$ .
Any subset $G$ of $Gal(K|k)$ gives a field $k \subseteq L \subseteq K$ , consisting of the elements of $K$ fixed by all the symmetries in $G$ .

or even like this:

Any set $k \subseteq L \subseteq K$ gives a subset $G$ of $Gal(K|k)$ , consisting of symmetries that fix every element of $L$ .
Any subset $G$ of $Gal(K|k)$ gives a set $k \subseteq L \subseteq K$ , consisting of the elements of $K$ fixed by all the symmetries in $G$ .

This could allow us to ‘discover’ the concepts of subgroup and intermediate field in the process of improving our Galois connection to a bijection, instead of putting in those concepts from the start.

I think this would work. I haven’t carefully checked.

But I decided that enough readers would know a bit about Galois theory already that the unfamiliarity of this new approach would create more confusion than enlightenment!

Posted by: John Baez on April 20, 2023 5:34 PM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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What is it when you direct readers off to Wikipedia, rather than to the $n$ Lab? It feels like someone insisting on calling up Pizza Hut for a delivery rather than asking for a hand-crafted pizza from your friends’ local start-up.

Of course, it’s possible you simply prefer the former.

Posted by: David Corfield on April 20, 2023 11:58 AM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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You’ve repeatedly criticized me for linking to Wikipedia instead of the nLab, and I keep trying to explain why I sometimes do that. Let me try again.

You’ll notice that in this article I carefully avoided any requirement that the readers already understand anything about category theory, Galois theory or algebraic geometry. I assumed only that readers know what a field is, what a group is, and that they can imagine a field has a group of “symmetries”. I didn’t say “automorphisms”, because that word, while more precise, is more scary. I didn’t assume they know what $k^n$ or $k[x_1, \dots, x_n]$ means. I explained those things. I didn’t assume they know what “ideal” means: I defined that term.

If you didn’t notice these things, maybe that’s good: I try to write in a way that works on several levels. But they’re easy to spot if you look.

So: this article was intended to be friendly for beginners. And the reason is that getting interested in Galois connections, and analogies between different subjects, is a great way for beginners to get interested in category theory.

When explaining math online, I choose my links based on the audience I have in mind. In many cases I feel the Wikipedia article offers a more gentle introduction to a topic than the corresponding nLab article. In every case I examine the relevant articles and choose the one I think is best for the purpose at hand: not the one I would benefit from most, but the one my intended audience would benefit from.

The Wikipedia article “Galois correspondence” starts:

In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois.

The nLab article “Galois correspondence” starts:

In order theory the term Galois connection (due to Ore [44], who spelled it â€œconnexionâ€) can mean both: â€œadjunction between posetsâ€ and â€œdual adjunction between posetsâ€; the former notion is sometimes called â€œmonotone Galois connectionâ€ and the latter â€œantitone Galois connectionâ€. In this article the term â€œGalois connectionâ€ shall mean â€œdual adjunction between posetsâ€.

I decided the former article helps the beginner more — and not just in this first paragraph, but throughout.

Posted by: John Baez on April 20, 2023 5:16 PM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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Just chiming in to say that, indeed, sometimes nLab is really daunting when you appreciate reading n-Cat Café but do not have more than a bachelor in maths.

Posted by: Malo Tarpin on April 21, 2023 8:48 AM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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Ah, OK, sorry for suggesting it wasn’t thought through on your part.

Of course, this raises the question of whether the $n$ Lab entries could be made gentler. Nobody has ever made any objection to the inclusion of pedagogical motivation. Indeed, some of my favourite pages are of this kind, e.g., motivation for sheaves, cohomology and higher stacks.

In an ideal world we’d run many more such pages. So in this case, it could result in a page on Galois connections, etc., which is more accessible than the wikipedia page.

But time is short!

Posted by: David Corfield on April 21, 2023 10:41 AM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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I spend about an hour day on average explaining math, science and more recently music theory online — for years I did it on Google+ and then Twitter, but now on Mathstodon. I need to do it to organize my thoughts and experience the pleasure of really understanding things. For example I just spent 45 minutes writing about the Lydian dominant scale.

If I felt sure my explanations would persist on Wikipedia or the nLab, I’d spend more time on those sites… but since they’re collaborative, I can’t count on my particular style persisting after several layers of subsequent revisions. For example, the nLab has a tendency to start explanations at the highest known level of generality, and then work down to special cases — the nPOV tends to push in this direction — while I prefer to start with elementary things and work up.

Since I want my explanations to stick around, I copy them to either my online diary, the n-Category Café and/or Azimuth. This proved wise after Google+ went bust and Twitter became insufferably antic.

Posted by: John Baez on April 21, 2023 6:24 PM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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I think this is a great kind of approach pedagogically. Something philosophically similar is often done in one of the fields I work in the vicinity of, namely high-dimensional convex geometry. A typical result in that field says something like $F(K) \le c(n) G(K)$ for every $n$ -dimensional convex body $K$ . Here $F$ and $G$ are geometric functionals of some kind, and $c(n)$ is a constant that depends only on the dimension $n$ .

Results like this are often trivial if you only want to know that some such $c(n)$ exists, without an explicit expression for it, but are usually also quite elementary (bargain basement) for a sufficiently large $c(n)$ (say, $c(n) = 2^n$ ). When someone is presenting a proof that a smaller $c(n)$ also works (say, $c(n) = n$ ), it can be quite enlightening to first see the cheap proof for the big $c(n)$ , and have it pointed out how and why the new proof does better. Not every speaker takes that approach, but I always appreciate the ones who do.

Posted by: Mark Meckes on April 21, 2023 2:58 PM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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Yes, I agree! These things are almost always enlightening and pleasant. On his blog my friend Michael Weiss has a link to a nice pedagogical note where he proves some results similar to Stirling’s formula — weaker, but much easier to prove.

Posted by: John Baez on April 21, 2023 11:12 PM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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Also, for many purposes those “weaker” results may be all that’s needed. In my own research and teaching I usually only need Weiss’s first and second approximations for Stirling’s formula, and I don’t think I ever wanted something sharper than his fifth approximation.

But I put those scare quotes up there because these easier results aren’t actually weaker. Unlike the usual asymptotic statement of Stirling’s formula, these approximations give concrete inequalities that hold for every $n$ . Which is vital for applications in geometry, statistics, computer science, etc., where $n$ represents something that is typically large, but always finite.

All of which is just to further emphasize that we should never turn up our noses at “simpler” versions of fancy theorems.

Posted by: Mark Meckes on April 25, 2023 5:13 PM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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I feel sure that if maths were taught more often in the way Prof Baez teaches it then there would be more maths lovers in the world!

Posted by: Bertie on April 24, 2023 12:12 AM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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The first couple of times I taught Galois theory, I explicitly included the bargain basement fundamental theorem, as a prelude to the full-on version. There’s definitely an argument for getting the most trivial stuff out of the way first.

Actually, there’s another bargain basement version of the fundamental theorem of Galois theory, namely, the case where both fields are finite. Then all the complicated stuff about separability and normality, which can seem so technical at least at first, vanishes into thin air.

In the same vein, there’s a bargain basement version of Fourier theory: the case of finite abelian groups. Then all the analytical complications (e.g. with different modes of convergence) vanish into thin air.

When I used to teach Fourier theory, I’d do the finite abelian group stuff at the end, as a kind of dessert or reward for making it through the tough stuff. Now I’m teaching Galois theory, I do the finite fields stuff at the end for similar reasons. In neither case did it occur to me to do them as a warm-up at the start. Too late now! This is my last year of Galois.

But I think the examples in your post are deeper discounts, or a deeper level of basement, than the ones I’ve just mentioned.

Posted by: Tom Leinster on May 1, 2023 9:49 PM | Permalink | Reply to this

Re: Bargain-Basement Mathematics

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Here are a few set theory results that may count as bargain basement. The flavour is a bit different from your examples, John, but here goes anyway.

König’s theorem states that whenever $(X_i)_{i \in I}$ and $(Y_i)_{i \in I}$ are families of sets satisfying $X_i \lt Y_i$ for all $i$ , then $\sum X_i \lt \prod Y_i$ . If you assume the generalized continuum hypothesis, this is almost obvious. For GCH says “ $A \lt B \iff 2^A \leq B$ ”, so König’s theorem becomes the statement that if $2^X_i \leq Y_i$ for all $i$ then $2^{\sum X_i} \leq \prod Y_i$ , which is easy. And I think this little argument actually gives some insight into why König’s theorem is the way it is, with a sum on one side and a product on the other.

There are probably lots of set theory results that are easy when you assume GCH and hard when you don’t. So, GCH is a common route to the bargain basement — a trade discount card?
Suppose you — an imaginary student — are convinced that $\mathbb{N} + \mathbb{N} \cong \mathbb{N}$ , but now I want to convince you that $X + X \cong X$ for all infinite $X$ . Well, that takes a bit of work. But if you’re willing to believe that any infinite set is of the form $\mathbb{N} \times (something)$ , then it’s easy, by distributivity of $\times$ over $+$ .
Suppose you’re now convinced that $X + X \cong X$ for all infinite $X$ , but next I want to persuade you that $Y \times Y \cong Y$ for all infinite $Y$ . Again, the full version takes some work. But I can easily persuade you in the case where $Y$ is a power set $2^X$ , since $2^X \times 2^X \cong 2^{X + X} \cong 2^X$ .

Posted by: Tom Leinster on May 1, 2023 10:02 PM | Permalink | Reply to this

König’s inequality

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In certain parts of model theory, it’s often convenient to assume GCH. As Sacks remarks in his Saturated Model Theory (an oldie but a goody), you can often get away with initially assuming GCH and then using absoluteness results to eliminate the assumption.

(My impression is that this approach has become somewhat obsolete, thanks to things like replendency.)

The König inequality has a curious history. König first stated it (or actually a weaker version) in 1904, at a lecture at the Third International Congress. He used it in an attempted proof that the reals could not be well-ordered. (This was before Zermelo’s proof of the well-ordering theorem.) Of course, it would follow that $2^{\aleph_0}$ is not an aleph, and so CH would be false in spades.

You’ll find the full story in Moore’s Zermelo’s Axiom of Choice (pp. 86-88), but briefly:

Suppose $2^{\aleph_0}=\aleph_\beta$ . In the inequality, set $|X_i|=\aleph_{\beta+i}$ and $|Y_i|=\aleph_{\beta+i+1}$ , getting

(1)

\aleph_{\beta+\omega}\lt{\aleph_{\beta+\omega}}^{\aleph_0}

But Bernstein (of the Schröder-Bernstein theorem) had supposed proved in his thesis,

(2)

{\aleph_\alpha}^{\aleph_0}=\aleph_\alpha\cdot 2^{\aleph_0}

for every $\alpha$ . So plugging in $\alpha=\beta+\omega$ and $2^{\aleph_0}=\aleph_\beta$ , we have

(3)

{\aleph_{\beta+\omega}}^{\aleph_0} =\aleph_{\beta+\omega}\cdot\aleph_\beta=\aleph_{\beta+\omega}

contradiction.

It turns out that Bernstein’s proof doesn’t work when $\alpha$ is a limit ordinal. Zermelo found the problem the next day. Later on Hausdorff did a careful analysis, uncovering the notions of cofinality and the distinction between regular and singular cardinals.

This episode was what stimulated Zermelo to work on the well-ordering problem in the first place. Altogether a very fruitful mistake!

(Incidentally, König continued to doubt that the reals could be well-ordered, and later came up with another “proof” that they couldn’t, this time using a variant of Berry’s paradox.)

Posted by: Michael Weiss on May 2, 2023 6:35 PM | Permalink | Reply to this

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April 19, 2023