### Bargain-Basement Mathematics

#### Posted by John Baez

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.

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?

## Re: Bargain-Basement Mathematics

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.)