
This week I'd like to say more about number theory, but first  here's the most fun book on astronomy I've ever seen:
1) James B. Kaler, The Hundred Greatest Stars, Copernicus Books (Springer Verlag), New York, 2002.
It's just what the title says: a compilation of the author's 100 favorite stars, each with a picture and a onepage description of what makes that star interesting. They're incredibly diverse, from the mammoth Eta Carinae to tiny brown dwarf Gliese 229B. You'll see soft gamma repeaters, yellow hypergiants, pulsars, Miratype variables, barium stars, symbiotic stars, and more. There's also an introduction that explains the concepts needed to enjoy all these different kinds of stars, like the HertzsprungRussell diagram and a bit of nuclear physics. With the help of this, the whole book forms a wonderful taxonomy of stellar astrophysics. I suggest keeping it by your bed and reading one star a night  though you may wind up staying up late and devouring the whole book.
On to number theory....
There's a widespread impression that number theory is about numbers, but I'd like to correct this, or at least supplement it. A large part of number theory  and by the far the coolest part, in my opinion  is about a strange sort of geometry. I don't understand it very well, but that won't prevent me from taking a crack at trying to explain it....
The basic idea is to push the analogy between integers and polynomials as far as it will go. They're similar because you can add, subtract and multiply them, and these operations satisfy the usual rules we all learned in high school:
x + y = y + x (x + y) + z = x + (y + z) x + 0 = x x + (x) = 0 xy = yx (xy)z = x(yz) x1 = x x(y + z) = xy + xzAnything satisfying these rules is called a "commutative ring". There are also a lot of deeper similarities between integers and polynomials, which I'll talk about later. But, there's a big difference! Polynomials are functions on the line, whereas the integers aren't functions on some space  at least, not in any instantly obvious way.
The fact that polynomials are functions on a space is what lets us graph them. This lets us think about them using geometry  and also think about geometry using them. This was the idea behind Descartes' "analytic geometry", and it was immensely fruitful.
So, it would be cool if we could also think about the integers using geometry. And it turns out we can, but only if we stretch our concept of geometry far enough!
If we do this, we'll see some cool things. First of all, we'll see that algebra is just like geometry, only backwards.
What do I mean by this? Well, whenever you have a map T: X → Y going from the space X to the space Y, you can use it to take functions on Y and turn them into functions on X. Since this goes backwards, it's called "pulling back along T". Here's how it goes: if f is a function on Y, we get a function T*(f) on X given by:
T*(f)(x) = f(T(x))Moreover, functions on a space form a commutative ring, since you can add and multiply them pointwise, and pulling back is a "homomorphism", meaning that it preserves all the structure of a commutative ring:
T*(f + g) = T*(f) + T*(g) T*(0) = 0 T*(fg) = T*(f) T*(g) T*(1) = 1Conversely, any sufficiently nice homomorphism from functions on Y to functions on X will come from some map T: X → Y this way! Here I'm summarizing a whole bunch of different theorems, each of which goes along with its own precise definition of "space", "map", and "nice".
Some of these theorems are technical, but the basic idea is simple: we can translate back and forth between the study of commutative rings (algebra) and the study of spaces (geometry) and by thinking of commutative rings as consisting of functions on spaces. We get a little dictionary for translating between geometry and algebra, like this:
GEOMETRY ALGEBRA spaces commutative rings maps homomorphismsBut be careful: this translation turns maps into homomorphisms going backwards: it's "contravariant". This is really important in two ways. First, suppose we have a point x in a space X. This gives a map
i: {x} → XThis, in turn, gives a homomorphism i* sending functions on X to functions on {x}. Functions on a onepoint space are like numbers, so i* acts like "evaluation at x". Moreover, i* will tend to be onto: that's the backwards analogue of the fact that i is onetoone!
Second, suppose we have a map from a space E onto the space X:
p: E → X.If you know some topology, think of E as a "covering space" of X. Then we get a homomorphism p* from functions on X to functions on E. Moreover p* will tend to be onetoone: that's the backwards version of the fact that p was onto!
We can use these examples to figure out the analogue of a "point" or a "covering space" in the world of commutative rings! And the resulting ideas turn out to be crucial to modern number theory.
In "week199" I explained the analogue of a "point" for commutative rings: it's a "prime ideal". So, now I want to explain the analogue of a "covering space". This will expand our dictionary so that it relates Galois groups to fundamental groups of topological spaces... and so on.
But, we won't get too far if we don't remember why a "prime ideal" is like a "point"! So, I guess I'd better review some of "week199" before charging ahead into the beautiful wilderness.
What's special about the ring of functions on a space consisting of just one point? Take real or complexvalued functions, for example. How do these differ from the functions on a space with lots of points?
The answer is pretty simple: on a space with just one point, a function that vanishes anywhere vanishes everywhere! So, the only function that fails to have a multiplicative inverse is 0. For bigger spaces, this isn't true.
A commutative ring where only 0 fails to have a multiplicative inverse is called a "field". So, the algebraic analogue of a onepoint space is a field.
This means that the algebraic analogue of a map from a onepoint space into some other space:
i: {x} → Xshould be a homomorphism from a commutative ring R to a field k:
f: R → k
Our translation dictionary now looks like this:
GEOMETRY ALGEBRA spaces commutative rings maps homomorphisms onepoint spaces fields maps from onepoint spaces homomorphisms to fieldsIt's worth noting some subtleties here. In the geometry we learned in high school, once we see one point, we've seen 'em all: all onepoint spaces are isomorphic. But not all fields are isomorphic! So, if we're trying to think of algebra as geometry, it's a funny sort of geometry where points come in different flavors!
Moreover, there are homomorphisms between different fields. These act like "flavor changing" maps  maps from a point of one flavor to a point of some other flavor.
If we have a homomorphism f: R → k and a homomorphism from k to some other field k', we can compose them to get a homomorphism f ': R → k'. So, we're doing some funny sort of geometry where if we have a point mapped into our space, we can convert it into a point of some other flavor, using a "flavor changing" map.
Now let's take this strange sort of geometry really seriously, and figure out how to actually turn a commutative ring into a space! First I'll describe what people usually do. Eventually I'll describe what perhaps they really should do  but maybe you can guess before I even tell you.
People usually cook up a space called the "spectrum" of the commutative ring R, or Spec(R) for short. What are the points of Spec(R)? They're not just all possible homomorphisms from R to all possible fields. Instead, we count two such homomorphisms as the same point of Spec(R) if they're related by a "flavor changing process". In other words, f ': R → k' gives the same point as f: R → k if you can get f ' by composing f with a homomorphism from k to k'.
This is a bit ungainly, but luckily there's a quick and easy way to tell when f: R → k and f ': R → k' are related by such a flavor changing process, or a sequence of such processes. You just see if they have the same kernel! The "kernel" of f: R → k is the subset of R consisting of elements r with
f(r) = 0The kernel of a homomorphism to a field is a "prime ideal", and two homomorphisms are related by a sequence of flavor changing processes iff they have the same kernel. Furthermore, every prime ideal is the kernel of a homomorphism to some field. So, we can save time by defining Spec(R) to be the set of prime ideals in R.
For completeness I should remind you what a prime ideal is! An "ideal" in a ring R is a set closed under addition and closed under multiplication by anything in R. It's "prime" if it's not all of R, and whenever the product of two elements of R lies in the ideal, at least one of them lies in the ideal.
So, we have something like this:
GEOMETRY ALGEBRA spaces commutative rings maps homomorphisms onepoint spaces fields maps from onepoint spaces homomorphisms to fields points of a space prime ideals of a commutative ringNow let's use these ideas to study "branched covering spaces" and their analogues in algebra. This week I'll talk about two examples. The first is very geometrical, and it should be familiar to anyone who has studied a little complex analysis. The second is more algebraic, and it's important in number theory. But, the cool part is that they fit into the same formalism!
If you don't know what a branched covering space is, don't worry: we'll start with the very simplest example. We'll look at this map from the complex plane to itself:
p: C → C p(z) = z^{2}Except for zero, every complex number has two square roots, so this map is twotoone and onto away from the origin. In fact, away from the origin you can visualize this thing locally as two sheets of paper sitting above one. But these two sheets have a global complication: if you start on the top sheet and hike once around the origin, you wind up on the bottom sheet  and vice versa! In topology we call this sort of thing a "double cover". When we include the point z = 0 things get even more complicated, since the two sheets meet there. So we have something trickier: a "branched cover". In general, a branched cover is like a covering space except that the different "sheets" can merge together at certain points, called "branch points".
Now let's think about this algebraically. To keep from getting confused, let's write
z^{2} = w
so that p is a map from the "z plane" down to the "w plane", sending each point z to the point z^{2} = w. The ring of polynomial functions on the z plane is called C[z]; the ring of polynomial functions on the w plane is called C[w]. We can pull functions from the w plane back up to the z plane:
p*: C[w] → C[z]
and p* works in the obvious way, taking any function f(w) to the function f(z^{2}).
Just as p is onto, p* is onetoone! So, we can think of C[w] as sitting inside C[z], consisting of those polynomials in z that only depend on z^{2}: the even functions. We say C[w] is a "subring" of C[z], or equivalently, that C[z] is an "extension" of C[w].
In this example we can get the bigger ring from the smaller one by throwing in solutions of some polynomial equations, so we call it an "algebraic extension". We've already seen some algebraic extensions, namely algebraic number fields, where take the field of rational numbers and throw in some solutions of polynomial equations. Algebraic extensions can be complicated, but this one is really simple: we just start with C[w] and throw in the solution of one polynomial equation, namely
z^{2} = wIt turns out that quite generally, algebraic extensions of commutative rings act a lot like branched covering spaces. I probably don't have the technical details perfectly straight, but let's add this to our translation dictionary, because it's an important idea:
GEOMETRY ALGEBRA spaces commutative rings maps homomorphisms onepoint spaces fields maps from onepoint spaces homomorphisms to fields points of a space prime ideals of a commutative ring branched covering spaces algebraic extensions of commutative ringsNow let's have some fun: let's see how our algebraic concept of "point", namely "prime ideal", interacts with our branched double cover of the complex plane. There's something straightforward going on, but also something more subtle and interesting.
The straightforward thing is that any point up on the z plane maps to one down on the w plane. We don't need fancy algebra to see this! But, it's worth doing algebraically. According to the fancy algebraic definition, a "point" in the spectrum of the commutative ring C[z] is a prime ideal. But as you might hope, these are the same as good oldfashioned points in the complex plane!
It works like this: given any point x in C, we get a homomorphism from C[z] to C called "evaluation at x", which sends any polynomial f to the number f(x). The kernel of this is the prime ideal consisting of all polynomials that vanish at x. These are just the polynomials containing a factor of z  x, so we call this ideal
<z  x>So, we get some prime ideals in C[z] from points of C this way. But in fact there's a theorem that every prime ideal in C[z] is of this form! So, we get a onetoone correspondence
Spec(C[z]) = CSimilarly,
Spec(C[w]) = CNow let's think about our branched cover
p: C → Cin different ways. It starts out life as a map from the z plane down to the w plane. We can use this to pull back functions on the w plane up to the z plane:
p*: C[w] → C[z]
But then, by general abstract baloney, the inverse image under p* of any prime ideal in C[z] is a prime ideal back in C[w]. This gives a map from Spec(C[z]) to Spec(C[w]). But this is just a map from the z plane to the w plane! And it's the same map p we started with. If you don't see why, it's a good exercise to check this.
So: we translated from geometry to algebra and back to geometry, and we got right back where we started. Note that each time we translated, our description of the map p got turned around backwards.
But there's a subtler and more interesting thing we can do with our branched cover. We can take a point down on the w plane and look at the points up on the z plane that map down to it!
Usually there will be two, but for the origin there's just one. This much is clear from thinking geometrically. But if we think algebraically, we'll see something funny going on at the origin. We can already see it geometrically: the origin is where the two sheets of our branched cover meet, so we call it a "branch point". But the algebraic viewpoint sheds an interesting new light on this.
What we'll do is take a prime ideal in C[w] and push it forwards via
p*: C[w] → C[z]The resulting subset won't be an ideal, but it will "generate" an ideal, meaning we can take the smallest ideal containing it. This ideal won't be prime, but we can "factor" it into prime ideals: there's a fairly obvious way to multiply ideals, and we happen to be working with rings where there's a unique way to factor any ideal into prime ideals.
Let's try it. First pick a number x that's not zero. It gives a prime ideal in C[w], namely
<w  x>Next push this ideal forwards via p* and let it generate an ideal in C[z], namely
<z^{2}  x>This is not prime, but we can factor it, which in this case simply amounts to factoring the polynomial that generates it:
<z^{2}  x> = <(z  sqrt(x)) (z + sqrt(x))> = <z  sqrt(x)> <z + sqrt(x)>We get a product of two prime ideals, corresponding to two points in the z plane, namely +sqrt(x) and sqrt(x). These are the two points that map down to x.
In this sort of situation, we say the prime ideal <w  x> "splits" into the prime ideals <z  sqrt(x)> and <z + sqrt(x)> when we go from C[w] to the extension C[z]. This is just an overeducated way of saying the number x has two different square roots.
But suppose x = 0. This doesn't have two square roots! Everything works the same except we get
<z^{2}> = <z> <z>We say the prime ideal <w> "ramifies" when we go from C[w] to the extension C[z]. We still get a product of prime ideals; they just happen to be the same. This is a way of making sense of the funny notion that the number 0 has two square roots... which just happen to be the same! Lots of mathematicians and physicists talk about "repeated roots" when an equation has "two solutions that just happen to be equal". This is just a way of making that precise.
But all this algebraic machinery must seem like overkill if this is the first time you've seen it. It pays off when we get to more algebraic examples. So, let me sketch the simplest one.
Let Z be the ring of integers, and let Z[i] be the ring of Gaussian integers, namely numbers of the form a+bi where a and b are integers. Z[i] is an algebraic extension of Z, since we can get it by throwing in a solution z of the polynomial equation
z^{2} = 1This equation is quadratic, just like it was in the example we just did! Now we're throwing in a square root of 1 instead of a square root of some function on the complex plane. But if we take the analogy between geometry and algebra seriously, this extension should still give some sort of "branched double cover"
p: Spec(Z[i]) → Spec(Z)What's this like?
It's actually really interesting, but I'll just sketch how it works.
The points of Spec(Z) are prime ideals in Z. In "week199" we saw that except for the prime ideal <0>, these are generated by prime numbers.
Similarly, except for <0>, the prime ideals in Z[i] are generated by "Gaussian primes": Gaussian numbers that have no factors except themselves and the "units" 1, 1, i and i. (A "unit" in a ring is an element with a multiplicative inverse; we don't count units as primes.)
The map p sends each Gaussian prime to a prime, and it's fun to work out how this goes... but it's even more fun to work backwards! Let's take primes in the integers and see what happens when we let them generate ideals in the Gaussian integers! This is like taking points in the base space of a branched cover and seeing what's sitting up above them.
For example, the prime 5 "splits". It has two prime factors in the Gaussian integers:
5 = (2 + i)(2  i)so in Z[i] the ideal it generates is a product of two prime ideals:
<5> = <2 + i> <2  i>This means that two different points in Spec(Z[i]) map down to the point <5> in Spec(Z), namely <2 + i> and <2  i>. So we indeed have something like a double cover!
On the other hand, the prime 2 "ramifies". It has two prime factors in the Gaussian integers:
2 = (1 + i)(1  i)but these two Gaussian primes generate the same prime ideal:
<1 + i> = <1  i>since if we multiply 1+i by the unit i we get 1i. So, in the Gaussian integers we have
<2> = <1 + i> <1 + i>A repeated factor! This is just what happened to the branch point in our previous example: it had "two points sitting over it, which happen to be the same".
So far, everything seems to be working nicely. But, besides splitting and ramification, there's a third thing that happens here, which didn't happen in our example involving the complex plane. In fact, this third option never happens when we're doing algebraic geometry over the complex numbers!
Here's how it works. Consider the prime 3. This is still prime in the Gaussian integers! It doesn't split, and it doesn't ramify. If we factorize the ideal generated by 3 in Z[i] we just get
<3> = <3>It doesn't do anything  it just sits there! So, we say this prime is "inert".
This may seem boring, but it's actually mysterious  and downright MADDENING if we take the analogy between geometry and algebra seriously. It's weird enough to have a "branched" cover where sheets merge at certain points, but at least in that case we can see they've merged: a prime ideal in our subring generates an ideal in the extension that's not prime, but is a product of several prime factors, some of which happen to be the same. But when a prime ideal in our subring generates a prime ideal in the extension, it's as if our "cover" has just one sheet over this point in the base space! And if this happens for a quadratic extension  as it just did  something seems to have gone horribly wrong with the nice idea that "quadratic extensions are like branched double covers".
Luckily, this puzzle has a nice resolution. We shouldn't have decategorified! When we started discussing "points" for a commutative ring, we saw they form a category in a nice way: there are points of different "flavors", with "flavorchanging operations" going between them. Then we freaked out and turned this category into a set by decreeing that two point are the same whenever there's a morphism between them. If we hadn't done this, we'd have seen more clearly how "inert" primes fit into a nice pattern along with "split" and "ramified" ones.
I'll probably talk about this more sometime, and also look more carefully at what happens to all the different primes when we go to the Gaussian integers  to show you that we are, indeed, doing number theory!
But for now, I just want to make a few comments about this idea of points of different "flavors".
In fact Grothendieck proposed an even more general idea of this sort in his second approach to "schemes", which is simpler but much less widely discussed than his first approach. Basically, he said that given a commutative ring R, we should not only consider points that are homomorphisms from R to any field, but also to any commutative ring. For each commutative ring k we get a set consisting of all "kpoints" of R, namely homomorphisms
f: R → kAnd, for each homomorphism g: k → k' we get a "flavor changing operation" that sends kpoints to k'points. So, we get a functor from CommRing to Set! He called such a functor a "scheme". We can get schemes from commutative rings as just described  these are called "affine schemes"  but there are also others, for example those coming from projective varieties.
Anyway, here are some places to read more about number theory... mostly with an emphasis on the geometric viewpoint and the issue of "splitting, ramification and inertia".
For a really quick and friendly nononsense introduction, try this:
2) Harold M. Stark, Galois Theory, Algebraic Number Theory, and Zeta Function, in From Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin, 1992, pp. 313393.
To dig a lot deeper, try this book by Neukirch:
3) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher, Springer, Berlin, 1986.
I already mentioned it, but it's worth mentioning again, because it's pretty elementary, and very clear on the analogy between "function fields" (fields of functions on Riemann surfaces) and "number fields" (algebraic number fields).
This book by Borevich and Shafarevich doesn't make the analogy to geometry explicit:
4) Z. I. Borevich and I. R. Shafarevich, Number Theory, trans. Newcomb Greenleaf, Academic Press, New York, 1966.
However, it has a nice concept of a "theory of divisors" for a commutative ring  and if you know a bit about divisors from algebraic geometry, you'll see that this is secretly very geometrical! They show how to classify algebraic extensions of commutative rings using a theory of divisors, and show how to get a theory of divisors using "valuations". This manages to accomplish a lot of what other texts do using "adeles", without actually mentioning adeles. I find this instructive.
This book goes much further in the geometric direction, but still without introducing schemes:
5) Dino Lorenzini, An Invitation to Arithmetic Geometry, American Mathematical Society, Providence, Rhode Island, 1996.
It's really great  very pedagogical! It develops number fields and function fields in parallel. You'll need to be pretty comfy with commutative algebra to work all the way through it, though.
If you want to learn about schemes  not the kind I just talked about, just the usual sort, which still includes cool "spaces" like Spec(Z)  try these:
6) V. I. Danilov, V. V. Shokurov, and I. Shafarevich, Algebraic Curves, Algebraic Manifolds and Schemes, Springer, Berlin, 1998.
7) David Eisenbud and Joe Harris, The Geometry of Schemes, Springer, Berlin, 2000.
Schemes have a reputation for being scary, but both these books try hard to make them less so, including lots of actual pictures of things like Spec(Z[i]) sitting over Spec(Z).
To wrap things up, I just want to mention two papers on subjects I'm fond of....
In "week172" I discussed Tarski's "high school algebra problem". This asks whether every identity involving 1, +, x, and exponentials that holds in the positive natural numbers follows from the eleven we learned in high school:
x + y = y + x (x + y) + z = x + (y + z) xy = yx (xy)z = x(yz) 1x = x x^{1} = x 1^{x} = 1 x(y + z) = xy + xz x^{(y + z)} = x^{y} x^{z} (xy)^{z} = x^{z} y^{z} x^{yz} = (x^{y})^{z}The rules of this game allow only purely equational reasoning  not stuff like mathematical induction. The reason is that this is secretly a problem about "universal algebra" or "algebraic theories", as explained in "week200".
It turns out the answer is no! In fact there are infinitely many more independent identities! Here is the first one, due to Wilkies:
[(x + 1)^{x} + (x^{2} + x + 1)^{x}]^{y} [(x^{3} + 1)^{y} + (x^{4} + x^{2} + 1)^{y}]^{x} = [(x + 1)^{y} + (x^{2} + x + 1)^{y}]^{x} [(x^{3} + 1)^{x} + (x^{4} + x^{2} + 1)^{x}]^{y}I just found a paper, apparently written after "week172", which gives a very detailed account of this problem:
8) Stanley Burris and Karen Yeats, The saga of the high school identities, available at http://web.archive.org/web/20070212200835/http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PREPRINTS/saga.ps
It includes some new results, like the smallest known algebraic gadget satisfying all the high school identities but not Wilkies' identity  but also more interesting things that are a bit harder to describe.
Also, here's a cool paper relating some of Ramanujan's work to string theory:
9) Antun Milas, Ramanujan's "Lost Notebook" and the Virasoro Algebra, available as math.QA/0309201.
A lot of Ramanujan's weird identities turn out to be related to concepts from string theory, suggesting that he was born about a century too soon to be fully appreciated... but this paper tackles an identity of his that nobody had managed to explain using string theory before.
Addendum: Here's something a friend of mine wrote, and an expanded version of my reply.
By the way, I very much liked your explanation of points and prime ideals. Up until now I haven't seen a satisfactory explanation of why points correspond to prime rather than maximal ideals, and although I haven't completely digested what you wrote, it looks like it might do the job...Both here and in my discussion of spectra in "week199", I've been avoiding saying the things people usually say. People usually note that a maximal ideal is the same as the kernel of a homomorphism ONTO a field, while a prime ideal is the same as the kernel of a homomorphism ONTO an integral domain. (Recall that an integral domain is a commutative ring where xy = 0 implies that x or y is zero.) If we define the "points" of a commutative ring R to be its maximal or prime ideals, we can therefore think of these as the kernels of homomorphisms from R onto fields or integral domains.
However, defining points in terms of homomorphisms ONTO a given sort of commutative ring is rather irksome, because it doesn't tell us how points transform under homomorphisms of commutative rings, nor how they transform under the "flavorchanging operations" I was describing. The problem is that the composite of a homomorphism with an onto homomorphism needn't be onto!
So, what really matters is that a prime ideal is the same as the kernel of a homomorphism TO a field. To see how this follows from the usual story, note that any integral domain is contained in a field called its "field of fractions"  just as Z is contained in Q. Any homomorphism ONTO the integral domain thus becomes a homomorphism TO this field, with the same kernel. Conversely, any homorphism TO a field becomes a homomorphism ONTO its image, with the same kernel  and this image is always an integral domain.
Much later, in 2013, someone wrote:
Dear Professor Baez,I replied:In TWF 205 you discuss the puzzle of inert primes and indicate that this puzzle has a nice resolution in terms of decategorification. You also indicated then that you may talk further about this at a later date. By any chance did you return to this topic and if so might you be able to point me to the appropriate TWF (or other reference)?
My apologies for asking about a nine year old remark.
Sorry, I never wrote more about that. I sort of forget what I was talking about, because it comes from a time long ago when I was discussing number theory with James Dolan, and he understood this stuff much better than me. But I think I can remember a bit if I try...
[...] you discuss the puzzle of inert primes and indicate that this puzzle has a nice resolution in terms of decategorification.I remember that the resolution of the puzzle is to refrain from decategorifying.
In Grothendieck's approach, the "kpoints" of a ring R of algebraic numbers, for various choices of field k, are homomorphisms
R → k
These are the objects of a category, in the manner I described, with "flavorchanging operations" coming from homomorphisms k → k' as the morphisms.
If we restrict our attention to the case I was talking about, where R = Z[i] is the Gaussian integers and k is a finite field, the "inert" primes correspond to situations where a kpoint down in Z lifts to two different but isomorphic kpoints in Z[i]. When we decategorify, as we do in the usual approach, it looks like there's just one point in Z[i] sitting over this point in Z.
Let me warm up by considering the prime 5, which is not inert. The ideal in Z generated by 5 splits into two ideals in Z[i], <2  i> and <2 + i>. Correspondingly, there are two kpoints
Z[i] → F_{5}
one having <2  i> as its kernel (it sends i to 2), and having <2 + i> as its kernel (it sends i to 2). These kpoints are not isomorphic.
On the other hand, 3 is inert. The ideal in Z generated by 3 does not split in Z[i]. Now we can find a field k = F_{9} such that Z[i] has two different kpoints with the ideal <3> as kernel. One kpoint
Z[i] → F_{9}
sends i to one of the square roots of 1 in F_{9}, the other sends it to the other. (We have to go up to F_{9} since there's no square root of 1 in F_{3}.) However, these two kpoints are isomorphic, since there's an automorphism of F_{9} interchanging its two square roots of 1.
Moral: When we work with prime ideals as points instead of homomorphisms whose kernels are these prime ideals, it looks like there's just one point in Spec(Z[i]) sitting over the prime ideal <3>. But when we define points to be homomorphisms, we see Z[i] has two isomorphic points Z[i] → F_{9} sitting over the unique point Z → F_{9} whose kernel is that prime ideal.
© 2004 John Baez
baez@math.removethis.ucr.andthis.edu
