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Let {k} be a field, and let {E} be a finite extension of that field; in this post we will denote such a relationship by {k \hookrightarrow E}. We say that {E} is a Galois extension of {k} if the cardinality of the automorphism group {\mathrm{Aut}(E/k)} of {E} fixing {k} is as large as it can be, namely the degree {[E:k]} of the extension. In that case, we call {\mathrm{Aut}(E/k)} the Galois group of {E} over {k} and denote it also by {\mathrm{Gal}(E/k)}. The fundamental theorem of Galois theory then gives a one-to-one correspondence (also known as the Galois correspondence) between the intermediate extensions between {E} and {k} and the subgroups of {\mathrm{Gal}(E/k)}:

Theorem 1 (Fundamental theorem of Galois theory) Let {E} be a Galois extension of {k}.

  • (i) If {k \hookrightarrow F \hookrightarrow E} is an intermediate field betwen {k} and {E}, then {E} is a Galois extension of {F}, and {\mathrm{Gal}(E/F)} is a subgroup of {\mathrm{Gal}(E/k)}.
  • (ii) Conversely, if {H} is a subgroup of {\mathrm{Gal}(E/k)}, then there is a unique intermediate field {k \hookrightarrow F \hookrightarrow E} such that {\mathrm{Gal}(E/F)=H}; namely {F} is the set of elements of {E} that are fixed by {H}.
  • (iii) If {k \hookrightarrow F_1 \hookrightarrow E} and {k \hookrightarrow F_2 \hookrightarrow E}, then {F_1 \hookrightarrow F_2} if and only if {\mathrm{Gal}(E/F_2)} is a subgroup of {\mathrm{Gal}(E/F_1)}.
  • (iv) If {k \hookrightarrow F \hookrightarrow E} is an intermediate field between {k} and {E}, then {F} is a Galois extension of {k} if and only if {\mathrm{Gal}(E/F)} is a normal subgroup of {\mathrm{Gal}(E/k)}. In that case, {\mathrm{Gal}(F/k)} is isomorphic to the quotient group {\mathrm{Gal}(E/k) / \mathrm{Gal}(E/F)}.

Example 2 Let {k= {\bf Q}}, and let {E = {\bf Q}(e^{2\pi i/n})} be the degree {\phi(n)} Galois extension formed by adjoining a primitive {n^{th}} root of unity (that is to say, {E} is the cyclotomic field of order {n}). Then {\mathrm{Gal}(E/k)} is isomorphic to the multiplicative cyclic group {({\bf Z}/n{\bf Z})^\times} (the invertible elements of the ring {{\bf Z}/n{\bf Z}}). Amongst the intermediate fields, one has the cyclotomic fields of the form {F = {\bf Q}(e^{2\pi i/m})} where {m} divides {n}; they are also Galois extensions, with {\mathrm{Gal}(F/k)} isomorphic to {({\bf Z}/m{\bf Z})^\times} and {\mathrm{Gal}(E/F)} isomorphic to the elements {a} of {({\bf Z}/n{\bf Z})^\times} such that {a(n/m) = (n/m)} modulo {n}. (There can also be other intermediate fields, corresponding to other subgroups of {({\bf Z}/n{\bf Z})^\times}.)

Example 3 Let {k = {\bf C}(z)} be the field of rational functions of one indeterminate {z} with complex coefficients, and let {E = {\bf C}(w)} be the field formed by adjoining an {n^{th}} root {w = z^{1/n}} to {k}, thus {k = {\bf C}(w^n)}. Then {E} is a degree {n} Galois extension of {k} with Galois group isomorphic to {{\bf Z}/n{\bf Z}} (with an element {a \in {\bf Z}/n{\bf Z}} corresponding to the field automorphism of {k} that sends {w} to {e^{2\pi i a/n} w}). The intermediate fields are of the form {F = {\bf C}(w^{n/m})} where {m} divides {n}; they are also Galois extensions, with {\mathrm{Gal}(F/k)} isomorphic to {{\bf Z}/m{\bf Z}} and {\mathrm{Gal}(E/F)} isomorphic to the multiples of {m} in {{\bf Z}/n{\bf Z}}.

There is an analogous Galois correspondence in the covering theory of manifolds. For simplicity we restrict attention to finite covers. If {L} is a connected manifold and {\pi_{L \leftarrow M}: M \rightarrow L} is a finite covering map of {L} by another connected manifold {M}, we denote this relationship by {L \leftarrow M}. (Later on we will change our function notations slightly and write {\pi_{L \leftarrow M}: L \leftarrow M} in place of the more traditional {\pi_{L \leftarrow M}: M \rightarrow L}, and similarly for the deck transformations {g: M \leftarrow M} below; more on this below the fold.) If {L \leftarrow M}, we can define {\mathrm{Aut}(M/L)} to be the group of deck transformations: continuous maps {g: M \rightarrow M} which preserve the fibres of {\pi}. We say that this covering map is a Galois cover if the cardinality of the group {\mathrm{Aut}(M/L)} is as large as it can be. In that case we call {\mathrm{Aut}(M/L)} the Galois group of {M} over {L} and denote it by {\mathrm{Gal}(M/L)}.

Suppose {M} is a finite cover of {L}. An intermediate cover {N} between {M} and {L} is a cover of {N} by {L}, such that {L \leftarrow N \leftarrow M}, in such a way that the covering maps are compatible, in the sense that {\pi_{L \leftarrow M}} is the composition of {\pi_{L \leftarrow N}} and {\pi_{N \leftarrow M}}. This sort of compatibilty condition will be implicitly assumed whenever we chain together multiple instances of the {\leftarrow} notation. Two intermediate covers {N,N'} are equivalent if they cover each other, in a fashion compatible with all the other covering maps, thus {L \leftarrow N \leftarrow N' \leftarrow M} and {L \leftarrow N' \leftarrow N \leftarrow M}. We then have the analogous Galois correspondence:

Theorem 4 (Fundamental theorem of covering spaces) Let {L \leftarrow M} be a Galois covering.

  • (i) If {L \leftarrow N \leftarrow M} is an intermediate cover betwen {L} and {M}, then {M} is a Galois extension of {N}, and {\mathrm{Gal}(M/N)} is a subgroup of {\mathrm{Gal}(M/L)}.
  • (ii) Conversely, if {H} is a subgroup of {\mathrm{Gal}(M/L)}, then there is a intermediate cover {L \leftarrow N \leftarrow M}, unique up to equivalence, such that {\mathrm{Gal}(M/N)=H}.
  • (iii) If {L \leftarrow N_1 \leftarrow M} and {L \leftarrow N_2 \leftarrow M}, then {L \leftarrow N_1 \leftarrow N_2 \leftarrow M} if and only if {\mathrm{Gal}(M/N_2)} is a subgroup of {\mathrm{Gal}(M/N_1)}.
  • (iv) If {L \leftarrow N \leftarrow M}, then {N} is a Galois cover of {L} if and only if {\mathrm{Gal}(M/N)} is a normal subgroup of {\mathrm{Gal}(M/L)}. In that case, {\mathrm{Gal}(N/L)} is isomorphic to the quotient group {\mathrm{Gal}(M/L) / \mathrm{Gal}(N/L)}.

Example 5 Let {L= {\bf C}^\times := {\bf C} \backslash \{0\}}, and let {M = {\bf C}^\times} be the {n}-fold cover of {L} with covering map {\pi_{L \leftarrow M}(w) := w^n}. Then {M} is a Galois cover of {L}, and {\mathrm{Gal}(M/L)} is isomorphic to the cyclic group {{\bf Z}/n{\bf Z}}. The intermediate covers are (up to equivalence) of the form {N = {\bf C}^\times} with covering map {\pi_{L \leftarrow N}(u) := u^m} where {m} divides {n}; they are also Galois covers, with {\mathrm{Gal}(N/L)} isomorphic to {{\bf Z}/m{\bf Z}} and {\mathrm{Gal}(M/N)} isomorphic to the multiples of {m} in {{\bf Z}/n{\bf Z}}.

Given the strong similarity between the two theorems, it is natural to ask if there is some more concrete connection between Galois theory and the theory of finite covers.

In one direction, if the manifolds {L,M,N} have an algebraic structure (or a complex structure), then one can relate covering spaces to field extensions by considering the field of rational functions (or meromorphic functions) on the space. For instance, if {L = {\bf C}^\times} and {z} is the coordinate on {L}, one can consider the field {{\bf C}(z)} of rational functions on {L}; the {n}-fold cover {M = {\bf C}^\times} with coordinate {w} from Example 5 similarly has a field {{\bf C}(w)} of rational functions. The covering {\pi_{L \leftarrow M}(w) = w^n} relates the two coordinates {z,w} by the relation {z = w^n}, at which point one sees that the rational functions {{\bf C}(w)} on {L} are a degree {n} extension of that of {{\bf C}(z)} (formed by adjoining the {n^{th}} root of unity {w} to {z}). In this way we see that Example 5 is in fact closely related to Example 3.

Exercise 6 What happens if one uses meromorphic functions in place of rational functions in the above example? (To answer this question, I found it convenient to use a discrete Fourier transform associated to the multiplicative action of the {n^{th}} roots of unity on {M} to decompose the meromorphic functions on {M} as a linear combination of functions invariant under this action, times a power {w^j} of the coordinate {w} for {j=0,\dots,n-1}.)

I was curious however about the reverse direction. Starting with some field extensions {k \hookrightarrow F \hookrightarrow E}, is it is possible to create manifold like spaces {M_k \leftarrow M_F \leftarrow M_E} associated to these fields in such a fashion that (say) {M_E} behaves like a “covering space” to {M_k} with a group {\mathrm{Aut}(M_E/M_k)} of deck transformations isomorphic to {\mathrm{Aut}(E/k)}, so that the Galois correspondences agree? Also, given how the notion of a path (and associated concepts such as loops, monodromy and the fundamental group) play a prominent role in the theory of covering spaces, can spaces such as {M_k} or {M_E} also come with a notion of a path that is somehow compatible with the Galois correspondence?

The standard answer from modern algebraic geometry (as articulated for instance in this nice MathOverflow answer by Minhyong Kim) is to set {M_E} equal to the spectrum {\mathrm{Spec}(E)} of the field {E}. As a set, the spectrum {\mathrm{Spec}(R)} of a commutative ring {R} is defined as the set of prime ideals of {R}. Generally speaking, the map {R \mapsto \mathrm{Spec}(R)} that maps a commutative ring to its spectrum tends to act like an inverse of the operation that maps a space {X} to a ring of functions on that space. For instance, if one considers the commutative ring {{\bf C}[z, z^{-1}]} of regular functions on {M = {\bf C}^\times}, then each point {z_0} in {M} gives rise to the prime ideal {\{ f \in {\bf C}[z, z^{-1}]: f(z_0)=0\}}, and one can check that these are the only such prime ideals (other than the zero ideal {(0)}), giving an almost one-to-one correspondence between {\mathrm{Spec}( {\bf C}[z,z^{-1}] )} and {M}. (The zero ideal corresponds instead to the generic point of {M}.)

Of course, the spectrum of a field such as {E} is just a point, as the zero ideal {(0)} is the only prime ideal. Naively, it would then seem that there is not enough space inside such a point to support a rich enough structure of paths to recover the Galois theory of this field. In modern algebraic geometry, one addresses this issue by considering not just the set-theoretic elements of {E}, but more general “base points” {p: \mathrm{Spec}(b) \rightarrow \mathrm{Spec}(E)} that map from some other (affine) scheme {\mathrm{Spec}(b)} to {\mathrm{Spec}(E)} (one could also consider non-affine base points of course). One has to rework many of the fundamentals of the subject to accommodate this “relative point of view“, for instance replacing the usual notion of topology with an étale topology, but once one does so one obtains a very satisfactory theory.

As an exercise, I set myself the task of trying to interpret Galois theory as an analogue of covering space theory in a more classical fashion, without explicit reference to more modern concepts such as schemes, spectra, or étale topology. After some experimentation, I found a reasonably satisfactory way to do so as follows. The space {M_E} that one associates with {E} in this classical perspective is not the single point {\mathrm{Spec}(E)}, but instead the much larger space consisting of ring homomorphisms {p: E \rightarrow b} from {E} to arbitrary integral domains {b}; informally, {M_E} consists of all the “models” or “representations” of {E} (in the spirit of this previous blog post). (There is a technical set-theoretic issue here because the class of integral domains {R} is a proper class, so that {M_E} will also be a proper class; I will completely ignore such technicalities in this post.) We view each such homomorphism {p: E \rightarrow b} as a single point in {M_E}. The analogous notion of a path from one point {p: E \rightarrow b} to another {p': E \rightarrow b'} is then a homomorphism {\gamma: b \rightarrow b'} of integral domains, such that {p'} is the composition of {p} with {\gamma}. Note that every prime ideal {I} in the spectrum {\mathrm{Spec}(R)} of a commutative ring {R} gives rise to a point {p_I} in the space {M_R} defined here, namely the quotient map {p_I: R \rightarrow R/I} to the ring {R/I}, which is an integral domain because {I} is prime. So one can think of {\mathrm{Spec}(R)} as being a distinguished subset of {M_R}; alternatively, one can think of {M_R} as a sort of “penumbra” surrounding {\mathrm{Spec}(R)}. In particular, when {E} is a field, {\mathrm{Spec}(E) = \{(0)\}} defines a special point {p_R} in {M_R}, namely the identity homomorphism {p_R: R \rightarrow R}.

Below the fold I would like to record this interpretation of Galois theory, by first revisiting the theory of covering spaces using paths as the basic building block, and then adapting that theory to the theory of field extensions using the spaces indicated above. This is not too far from the usual scheme-theoretic way of phrasing the connection between the two topics (basically I have replaced étale-type points {p: \mathrm{Spec}(b) \rightarrow \mathrm{Spec}(E)} with more classical points {p: E \rightarrow b}), but I had not seen it explicitly articulated before, so I am recording it here for my own benefit and for any other readers who may be interested.

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As readers who have followed my previous post will know, I have been spending the last few weeks extending my previous interactive text on propositional logic (entitied “QED”) to also cover first-order logic.  The text has now reached what seems to be a stable form, with a complete set of deductive rules for first-order logic with equality, and no major bugs as far as I can tell (apart from one weird visual bug I can’t eradicate, in that some graphics elements can occasionally temporarily disappear when one clicks on an item).  So it will likely not change much going forward.

I feel though that there could be more that could be done with this sort of framework (e.g., improved GUI, modification to other logics, developing the ability to write one’s own texts and libraries, exploring mathematical theories such as Peano arithmetic, etc.).  But writing this text (particularly the first-order logic sections) has brought me close to the limit of my programming ability, as the number of bugs introduced with each new feature implemented has begun to grow at an alarming rate.  I would like to repackage the code so that it can be re-used by more adept programmers for further possible applications, though I have never done something like this before and would appreciate advice on how to do so.   The code is already available under a Creative Commons licence, but I am not sure how readable and modifiable it will be to others currently.  [Update: it is now on GitHub.]

[One thing I noticed is that I would probably have to make more of a decoupling between the GUI elements, the underlying logical elements, and the interactive text.  For instance, at some point I made the decision (convenient at the time) to use some GUI elements to store some of the state variables of the text, e.g. the exercise buttons are currently storing the status of what exercises are unlocked or not.  This is presumably not an example of good programming practice, though it would be relatively easy to fix.  More seriously, due to my inability to come up with a good general-purpose matching algorithm (or even specification of such an algorithm) for the the laws of first-order logic, many of the laws have to be hard-coded into the matching routine, so one cannot currently remove them from the text.  It may well be that the best thing to do in fact is to rework the entire codebase from scratch using more professional software design methods.]

[Update, Aug 23: links moved to GitHub version.]



Every four years at the International Congress of Mathematicians (ICM), the Fields Medal laureates are announced.     Today, at the 2018 ICM in Rio de Janeiro, it was announced that the Fields Medal was awarded to Caucher Birkar, Alessio Figalli, Peter Scholze, and Akshay Venkatesh.

After the two previous congresses in 2010 and 2014, I wrote blog posts describing some of the work of each of the winners.  This time, though, I happened to be a member of the Fields Medal selection committee, and as such had access to a large number of confidential letters and discussions about the candidates with the other committee members; in order to have the opinions and discussion as candid as possible, it was explicitly understood that these communications would not be publicly disclosed.  Because of this, I will unfortunately not be able to express much of a comment or opinion on the candidates or the process as an individual (as opposed to a joint statement of the committee).  I can refer you instead to the formal citations of the laureates (which, as a committee member, I was involved in crafting, and then signing off on), or the profiles of the laureates by Quanta magazine; see also the short biographical videos of the laureates by the Simons Foundation that accompanied the formal announcements of the winners. I am sure, though, that there will be plenty of other mathematicians who will be able to present the work of each of the medalists (for instance, there was a laudatio given at the ICM for each of the winners, which should eventually be made available at this link).

I know that there is a substantial amount of interest in finding out more about the inner workings of the Fields Medal selection process.  For the reasons stated above, I as an individual will unfortunately be unable to answer any questions about this process (e.g., I cannot reveal any information about other nominees, or of any comparisons between any two candidates or nominees).  I think I can safely express the following two personal opinions though.  Firstly, while I have served on many prize committees in the past, the process for the Fields Medal committee was by far the most thorough and deliberate of any I have been part of, and I for one learned an astonishing amount about the mathematical work of all of the shortlisted nominees, which was an absolutely essential component of the deliberations, in particular giving the discussions a context which would have been very difficult to obtain for an individual mathematician not in possession of all the confidential letters, presentations, and other information available to the committee (in particular, some of my preconceived impressions about the nominees going into the process had to be corrected in light of this more complete information).  Secondly, I believe the four medalists are all extremely deserving recipients of the prize, and I fully stand by the decision of the committee to award the Fields medals this year to these four.

I’ll leave the comments to this post open for anyone who wishes to discuss the work of the medalists.  But, for the reasons above, I will not participate in the discussion myself.

[Edit, Aug 1: looks like the ICM site is (barely) up and running now, so links have been added.  At this time of writing, there does not seem to be an online announcement of the composition of the committee, but this should appear in due course. -T.]

[Edit, Aug 9: the composition of the Fields Medal Committee for 2018 (which included myself) can be found here. -T.]