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In my recent number theory seminar on “Hilbert’s 90 and generalizations” (notes here), Professor Goins asked the following interesting question.

Let {K} be a field and {d\in K^*}. Define {T_d} to be the torus

\displaystyle \left\{ \begin{pmatrix} x & dy \\ y & x \end{pmatrix} : x,y \in L, x^2-dy^2=1\right\}.

What values of {d} give {K}-isomorphic tori?

(The question was perhaps motivated by the observation that over the reals, the sign of {d} determines completely whether {T_d} would be split (i.e., isomorphic to {\mathbb R^*}) or anisotropic (i.e., isomorphic to {S^1}).

Here are two ways of looking at the answer.

  • For {d,e \in K^*}, we determine when two matrices {\displaystyle\begin{pmatrix} x & dy \\ y & x \end{pmatrix}} and {\displaystyle\begin{pmatrix} u & ev \\ v & u \end{pmatrix}} are conjugate in {\text{SL}_2(K)}. Solving the system

\displaystyle \begin{pmatrix} x & dy \\ y & x \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} . \begin{pmatrix} u & ev \\ v & u \end{pmatrix} . \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

gives {\displaystyle \frac{e}{d} = \left(\frac{b}{c}\right)^2, de = \left(\frac{d}{a}\right)^2}.

Thus {e \in d.(K^*)^2} i.e., the {T_d}‘s are classified by {\displaystyle \frac{K^*}{(K^*)^2}}. (For {K=\mathbb R}, this is isomorphic to {\{\pm 1\}} so the sign of {d} determines {T_d} upto conjugation.) By Kummer theory, {\displaystyle \frac{K^*}{(K^*)^2} \cong H^1(\text{Gal}(\overline K / K), \mu_2)}, where {\mu_2 = \{\pm 1\}} are the second roots of unity. Thus there is a correspondence between isomorphism classes of tori {T_d \; (d \in K^*)} and quadratic extensions of {K}.

  • Another way to look at the same thing is as follows. Fix {d \in K^*}. Let {L} be an extension of {K} wherein {T_d} splits. Now {T_d(L)} is a split torus of rank 1. For an algebraic group {G} over an algebraically closed field, we have the exact sequence

\displaystyle 1 \rightarrow \text{Inn}(G) \rightarrow \text{Aut}(G) \rightarrow \text{Aut}(\Psi_0(G)) \rightarrow 1,

where {\Psi_0(G)} is the based root datum {(X,\Delta,X\;\check{}, \Delta\,\check{})} associated to {G}. (Here, {X = X^*(G)} and {\Delta} is the set of simple roots of {X} corresponding to a choice of a Borel subgroup of {G}.) For details, see Corollary 2.14 of Springer’s paper “Reductive Groups” in Corvallis.

In our case, {G = T_d} so {\Psi_0(G) = ( \mathbb Z, \emptyset, \mathbb Z \;\check{}, \emptyset)}.

\displaystyle \text{Aut}(\Psi_0(G)) \cong \text{Aut}(\mathbb Z) \cong \{ \pm 1\}.

Now {L/K} forms of {T_d} are in bijective correspondence with

\displaystyle H^1(\text{Gal}(L/K), \text{Aut}(\Psi_0(G))) = H^1(\text{Gal}(L/K), \{\pm 1\}) \cong \text{Hom}_{\mathbb Z}(\text{Gal}(L/K), \{\pm 1\});

the last isomorphism because the Galois group acts trivially on the split torus {T_d(L)}. {\blacksquare}

In their book, Singer & Thorpe say, “At the present time, the average undergraduate Mathematics major finds math heavily compartmentalized.” One learns many things but does not see the connections between seemingly different things. Indeed, as the great Poincare says, Mathematics is the art of giving the same name to different things. In this and the subsequent post, we shall see the connections in algebra and topology with respect the Galois theory.

Galois theory in Algebra

This topic is covered in most standard algebra texts. It deals with studying the roots of polynomials and their relations. Given a field F and an irreducible polynomial p(x) with coefficients in F, we look at the smallest field K containing F and which has all roots of p(x). The set of all permutations of the roots of p  correspond to automorphisms of K which fix F element-wise. These automorphisms form a group known as the Galois group. There is a beautiful correspondence between subgroups of this group and subfields of K fixing F.

The 19-th century mathematicians Galois and Abel studied this group and Galois came up with useful characterization on this group as to when its roots could be expressed in terms of the coefficients. Of course the theory has now been much more generalized, abstracted and is used indispensably in many parts of mathematics – number theory, algebraic geometry and more.

Galois theory in Topology

(Even this can be found in any standard text on algebraic topology. But here I am talking about Riemann surfaces – and the connections between the two Galois theories — the too-fascinating-to-be-true connection, although not very difficult, its not found in lower-level texts. I will explain that in a follow-up post to this soon).

Given a point P on a topological space S, one talks about the equivalence class (modulo continuous deformations) of paths starting and ending at P. They form a group, the fundamental group. Also, given two (path)-connected spaces R and S, one says that R is a covering of S if there is a continuous surjective map from R to S such that every point of S has a neighbourhood U whose inverse image is a disjoint union of copies each resembling U. One may imagine the real line to be a covering of the unit circle via the map t being mapped to (cos t, sin t).

Now comes the Galois correspondence. For a `nice’ topological space, there is a natural one-to-one correspondence between subgroups of the fundamental group of that space and its covering spaces (rather, isomorphism classes of covering spaces, to be pedantic). Further, the fundamental groups of these spaces have fundamental group isomorphic to the subgroup we started with!

More analogy

The analogy between Galois groups of algebraic objects and fundamental groups and covering spaces of topological spaces goes beyond just one-to-one correspondence. A field extension is normal if it has enough automorphisms. A covering map too is normal (or Galois) if it has enough automorphisms!

Normal field extension \longleftrightarrow Normal subgroups of the Galois group

Normal covering \longleftrightarrow Normal subgroup of the fundamental group

In the next post, we shall see a deeper connection between the four objects above. Namely, we shall take a polynomial, construct its Galois group, get a covering map for this field extension and see that the two groups are the same!

Galois group = fundamental group.

Amazing stuff!

About me

Abhishek Parab

I? An Indian. A mathematics student. A former engineer. A rubik's cube addict. A nature photographer. A Pink Floyd fan. An ardent lover of Chess & Counter-Strike.

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