Open questions from Serre's 1984 paper on the Witt invariant of the trace form: a totally real Monster extension of $\mathbb{Q}$

Serre's 1984 paper 'L'invariant de Witt de la forme Tr(x^2)' computes the Witt (Hasse-Witt) invariant $w_2(Q_E)$ of the trace form of an etale algebra, proving $w_2(Q_E)=e^*(s_n)+(2,d_E)$, and raises three explicit questions: (1) extend Witt's 1936 explicit construction of solutions of the double-cover embedding problem beyond the few cases Witt treated; (2) construct explicitly the quadratic extension with Galois group $2^+\mathfrak{S}_{18}$ over the Galois closure of a trinomial $X^{18}+aX+b$ with group $\mathfrak{S}_{18}$, for which Serre proved solvability; (3) decide whether $\mathbb{Q}$ admits a totally real Galois extension with group the Monster $F_1$, Thompson's rigidity construction yielding only imaginary extensions. Of the paper's three questions, only (3) remains open.

Background

Inverse Galois theory asks which finite groups $G$ occur as $\mathrm{Gal}(E/\mathbb{Q})$ for a finite Galois extension $E$ of the rationals. One construction principle for large simple groups is rigidity: choose a triple of conjugacy classes in $G$ whose structure constant is 1 (a rigid triple); the associated branched cover of the projective line is then defined over $\mathbb{Q}$, and specializations of it realize $G$ as a Galois group over $\mathbb{Q}$. Thompson used this in 1984, with the rigid triple of classes 2A/3B/29A (in ATLAS notation: the conjugacy class of involutions, a class of elements of order 3, and a class of elements of order 29), to construct the first Galois extensions of $\mathbb{Q}$ with group the Monster $F_1$ - the largest sporadic simple group, i.e. the largest of the 26 exceptional finite simple groups lying outside the infinite cyclic, alternating and Lie-type families - of order about $8\times10^{53}$; Serre's Bourbaki exposé 689 (1988) recounts the construction. All fields produced this way are imaginary.

For a finite etale algebra $E$ of rank $n$ over a field $K$ of characteristic not 2 (a product of finite separable field extensions; the case of one factor is a number field), the trace form $Q_E(x)=\mathrm{Tr}_{E/K}(x^2)$ is a rank-$n$ quadratic form over $K$. Its two basic invariants are the discriminant $d_E\in K^\times/K^{\times2}$ and the Witt invariant $w_2(Q_E)$, also called the Hasse-Witt invariant: the degree-2 invariant with values in the 2-torsion $\mathrm{Br}_2(K)$ of the Brauer group of $K$ (the group of similarity classes of finite-dimensional central simple $K$-algebras), computable from a diagonalization $Q_E\simeq\langle a_1,\dots,a_n\rangle$ as the product of quaternion symbols $(a_i,a_j)$ - the quaternion symbol $(a,b)$ being the Brauer class of the 4-dimensional algebra generated by elements $i,j$ with $i^2=a$, $j^2=b$, $ij=-ji$ - and the invariant governing the Clifford algebra (the associative algebra canonically attached to a quadratic form) of the form. When $E/K$ is Galois with group $G_E$, numbering the $n$ embeddings defines a 1-cocycle $e$ with values in the symmetric group $\mathfrak{S}_n$; pulling back the generator $s_n$ of the mod-2 cohomology $H^2(\mathfrak{S}_n,\mathbb{Z}/2)$ (the second Stiefel-Whitney class, a mod-2 characteristic class attached to an orthogonal or permutation representation, of the natural permutation representation of $\mathfrak{S}_n$) gives a class $e^*s_n\in\mathrm{Br}_2(K)$ independent of the numbering. Fröhlich (1985) developed the orthogonal-representation framework around these invariants. Serre's Theorem 1 (1984) relates the two sides: $w_2(Q_E)=e^*s_n+(2,d_E)$, where $(a,b)$ is the quaternion symbol over $K$.

Over $\mathbb{Q}$, writing $r_1+2r_2=n$ for the number of real embeddings and pairs of complex-conjugate embeddings of $E$ (so $E$ is totally real exactly when $r_2=0$), Serre's Proposition 3 states: $d_E=1$ and $e^*s_n=0$ if and only if $r_2\equiv 0\pmod 4$ and $Q_E$ is isomorphic to the diagonal form $Q_{r_1,r_2}$ with $r_1+r_2$ entries $+1$ and $r_2$ entries $-1$.

The class $e^*s_n$ is also the shadow of a classical embedding problem. Let $2^+\mathfrak{S}_n$, also written $\tilde{\mathfrak{S}}_n$, denote the non-split double cover of $\mathfrak{S}_n$ (one of the two Schur double covers, realized inside a spin group, the double cover of an orthogonal group acting on a Clifford algebra; there is an analogous cover $2^+G$ for subgroups $G\subseteq\mathfrak{S}_n$). Given $E/K$ with $G_E\subseteq\mathfrak{S}_n$, the embedding problem asks for a quadratic extension $\tilde{E}/E$ whose Galois group over $K$ is the preimage $2^+G$; solvability is controlled by the vanishing of the corresponding Brauer class. Witt (1936) gave an explicit construction in the cases then accessible, from an isomorphism between the forms $X^2+Y^2+Z^2$ and $xX^2+yY^2+zZ^2$; Serre recovers Witt's result in his §3.2 and asks for extensions of the construction. Serre's §3.3 determines $e^*s_n$ for algebras defined by trinomials $X^n+aX+b$ and finds it identically 0 for $n=18,19,50,51,98,\dots$ (the list as printed in the paper); for $n=18$ and $G_E=\mathfrak{S}_{18}$ the embedding problem is therefore solvable, and Serre asks how to construct such an extension effectively. His §3.4 notes that of the 26 sporadic simple groups, 17 - including the Monster - have Schur multiplier (the second homology group $H_2(G,\mathbb{Z})$, which governs the central extensions of $G$) of odd order and hence trivial mod-2 $H^1$ and $H^2$, so that $e^*s_n=0$ automatically, and then poses the question that is the subject of this task.

The totally real branch of inverse Galois theory is far less developed than the general problem: the first totally real polynomials with Galois group $\mathrm{PSp}(6,2)$ (the projective symplectic group over the field of two elements, of order about $1.5\times10^6$) were obtained only in 2020 through a computation of multi-branch-point Hurwitz families (moduli spaces of branched covers of the projective line with prescribed Galois group and branch-point count), and 2026 work realizes $\mathrm{SL}_2(\mathbb{F}_{13})$ and $\mathrm{SL}_2(\mathbb{F}_{19})$ totally real via solvable central embedding problems combined with quadratic twists (the standard operation producing from a given realization a quadratic deformation with the same Galois group but possibly different signature). No currently available method enforces $r_2=0$ for realizations produced by rigidity.

Problem Statement

Part (1) was settled in the later literature and part (2) in principle by a published effective method; the open question is:

Does there exist a totally real Galois extension $E/\mathbb{Q}$, that is, with $r_2=0$ (every embedding $E\hookrightarrow\mathbb{C}$ has real image), such that $\mathrm{Gal}(E/\mathbb{Q})$ is isomorphic to the Monster sporadic simple group $F_1$?

Equivalently: an irreducible polynomial with rational coefficients, all of whose roots are real, whose splitting field has Galois group isomorphic to $F_1$ (a totally real Galois extension equals its own Galois closure, so it is totally real exactly when its primitive element has all real roots; conversely the splitting field of an irreducible polynomial with all real roots is totally real, so the two formulations coincide). Thompson's rigidity construction over $\mathbb{Q}$ yields only imaginary $F_1$-extensions, and no invariant of the source paper obstructs the totally real case: $F_1$ has trivial mod-2 $H^1$ and $H^2$, so $e^*s_n=0$ automatically, and $r_2=0$ satisfies the necessary congruence $r_2\equiv 0\pmod 4$ of Proposition 3. The question must therefore be decided by a construction or an impossibility argument beyond the paper's methods; a proof that no such extension exists resolves it equally.

References

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