Self-dual normal bases for $G$-Galois algebras over fields of $2$-cohomological dimension at most $2$: does vanishing of the degree-one and degree-two classes from $G$ suffice?

A self-dual normal basis of a $G$-Galois algebra $L$ over a field $k$ of characteristic $\ne 2$ is a normal basis orthonormal for the trace form $(x,y)\mapsto\mathrm{Tr}_{L/k}(xy)$. Bayer-Fluckiger and Serre's 1994 paper "Torsions quadratiques et bases normales autoduales" gave this existence question a cohomological framework and left several directions open, most prominently whether their number-field criterion — for $H^1(G,\mathbb{Z}/2\mathbb{Z})=H^2(G,\mathbb{Z}/2\mathbb{Z})=0$ the only obstruction lies at real places — extends to all fields of $2$-cohomological dimension $\le 2$. The later literature resolves the headline extension: Bayer-Fluckiger and Parimala's 2017 degree-two invariants $c_A(L)$ and $d_A(L)$ give a complete criterion over such fields, and their Corollary 5.5 proves that $H^1(G)=H^2(G)=0$ forces a self-dual normal basis; Voevodsky's proof of the Milnor conjecture discharges the 1994 conditional on Pfister-form criteria beyond rank $4$; and local and global fields are completely settled. The surviving open core, isolated as an explicit question in the 2017 paper, is whether for every $G$-Galois algebra over a field with $2$-cohomological dimension $\le 2$ the vanishing of all classes $x_L=\varphi^*(x)$ with $x$ of degree one and two from $G$ already implies a self-dual normal basis. This is known when $\varphi:\Gamma_k\to G$ is surjective ($L$ a field) and fails over fields of $2$-cohomological dimension $3$. Secondary surviving directions are criteria beyond $2$-cohomological dimension $2$ and effective computation of the invariants.

Background

Let $k$ be a field of characteristic $\ne 2$ and $G$ a finite group. A $G$-Galois algebra over $k$ is a finite étale $k$-algebra $L$ together with an action of $G$ by $k$-automorphisms such that the fixed subalgebra is $k$ and $\dim_k L=|G|$; the prototypical examples are Galois field extensions $L/k$ with $\mathrm{Gal}(L/k)=G$ and products of these, and the class is stable under base change. The trace form of $L$ is the bilinear form $q_L(x,y)=\mathrm{Tr}_{L/k}(xy)$; it is a $G$-quadratic form, meaning $q_L(gx,gy)=q_L(x,y)$ for all $g\in G$. A normal basis of $L$ is a $k$-basis of the form $(gx)_{g\in G}$ for a single $x\in L$, and it is self-dual when $q_L(gx,hx)=1$ for $g=h$ and $0$ for $g\ne h$, i.e. when the basis is orthonormal for $q_L$; equivalently, $L$ has a self-dual normal basis (SDNB) exactly when the $G$-quadratic form $(L,q_L)$ is isomorphic to the unit $G$-form, the form carried by the group algebra $k[G]$. Which $G$-Galois algebras admit an SDNB was studied from the late 1980s — Bayer-Fluckiger and Lenstra settled the odd-degree side — and was given a cohomological framework by Bayer-Fluckiger and Serre in the 1994 paper "Torsions quadratiques et bases normales autoduales", the source of the present problem.

The framework is as follows. Let $\Gamma_k$ be the absolute Galois group of $k$. A $G$-Galois algebra corresponds to a continuous homomorphism $\varphi:\Gamma_k\to G$, well defined up to conjugacy, and for each $n\ge 1$ the pullback acts on group cohomology with trivial $\mathbb{Z}/2\mathbb{Z}$-coefficients, $\varphi^*:H^n(G)=H^n(G,\mathbb{Z}/2\mathbb{Z})\to H^n(k)=H^n(\Gamma_k,\mathbb{Z}/2\mathbb{Z})$; write $x_L=\varphi^*(x)$. Bayer-Fluckiger and Serre proved that $x_L=0$ for all $x\in H^1(G)$ is necessary for an SDNB (the $H^1$-condition; equivalently $\varphi(\Gamma_k)$ lies in $G^2$, the subgroup generated by the squares of $G$), and sufficient over fields of $2$-cohomological dimension $\le 1$. Here $\mathrm{cd}_2(\Gamma_k)$, the $2$-cohomological dimension, is the smallest $d$ with $H^i(\Gamma_k,C)=0$ for all $i>d$ and all finite $2$-primary $\Gamma_k$-modules $C$; finite fields have $\mathrm{cd}_2\le 1$, while nonarchimedean local fields, function fields of curves over finite fields, and totally imaginary number fields have $\mathrm{cd}_2=2$. Over number fields, their Théorème 3.2.1 states that if $H^1(G)=H^2(G)=0$ then the only obstruction to an SDNB is at real places, so that over totally imaginary number fields every such $G$-Galois algebra has an SDNB (Corollaire 3.2.2). For groups whose $2$-Sylow subgroup $S$ (a maximal $2$-subgroup) is elementary abelian of rank $r$ with transitive fusion (the normalizer of $S$ permutes the involutions of $S$ transitively), they reduced the existence question to a single degree-$r$ class $z_L$: after an odd-degree extension the trace form becomes $m$ copies of a form $q_L^\perp$ of rank $2^r$ with $|G|=2^r m$, and $2^r q_L^\perp$ is an $r$-fold Pfister form — a tensor product $\langle\!\langle a_1\rangle\!\rangle\otimes\cdots\otimes\langle\!\langle a_r\rangle\!\rangle$ with $\langle\!\langle a\rangle\!\rangle=\langle 1,-a\rangle$ — whose Arason invariant $e_r$ (the cup product $(a_1)\cdots(a_r)\in H^r(k)$) is computed from a fundamental class $z\in H^r(G)$ via $z_L$; the sufficiency of the resulting criterion $z_L=0$ required the Arason invariant to classify $r$-fold Pfister forms, then known for $r\le 4$ and conditionally on the Milnor conjectures (proved later by Voevodsky: they identify Milnor K-theory mod $2$ with Galois cohomology and imply that an $r$-fold Pfister form is hyperbolic as soon as its Arason invariant vanishes, for every $r$). Two of their counterexamples delimit the whole circle: over a field of $\mathrm{cd}_2=3$, all positive-degree classes $x_L$ can vanish while no SDNB exists; over $\mathbb{Q}$ an SDNB can exist while some degree-two class $x_L\ne 0$.

The decisive later development is the 2017 paper of Bayer-Fluckiger and Parimala, "Cohomological invariants for $G$-Galois algebras and self-dual normal bases". Let $k[G]_s$ be the semisimple quotient of the group algebra, $\sigma$ its canonical involution, and let $A$ run over the central simple factors stable under $\sigma$ that are orthogonal or unitary, with center $E_A$; for $H=\varphi(\Gamma_k)$ the image of the absolute Galois group, certain central extensions of $H$ by $\mathbb{Z}/2\mathbb{Z}$ have extension classes $e_A\in H^2(H)$, and the invariants are $c_A(L)=\varphi^*(e_A)$ in the orthogonal case and $d_A(L)=\varphi^*(e_A)$ in the unitary case, both in $H^2(k)$. Their Theorem 5.3 shows that over a field with $\mathrm{cd}_2(\Gamma_k)\le 2$, an SDNB exists if and only if the $H^1$-condition holds, $\mathrm{Res}_{E_A/k}(c_A(L))=0$ in the $2$-torsion $\mathrm{Br}_2(E_A)$ of the Brauer group of $E_A$ modulo the class of $A$ for all orthogonal factors, and $\mathrm{Res}_{E_A/k}(d_A(L))=0$ in $\mathrm{Br}_2(E_A)$ for all unitary factors. Corollary 5.5 draws the headline conclusion: if $\mathrm{cd}_2(\Gamma_k)\le 2$ and $H^1(G)=H^2(G)=0$, then every $G$-Galois algebra over $k$ has an SDNB — resolving the 1994 open problem (the 1994 proof for imaginary number fields was known to extend to all $\mathrm{cd}_2\le 2$ fields once combined with Bayer-Fluckiger–Parimala's 1995 structure theorems for hermitian forms over such fields, and the 2017 paper gives an alternative proof through the new invariants). Together with the Hasse principle of Bayer-Fluckiger–Parimala–Serre (2013) — over a global field an SDNB exists if and only if one exists everywhere locally — and the 2017 local and global criteria, the existence question is completely settled over local and global fields.

One precise gap remains, stated as an explicit question by Bayer-Fluckiger and Parimala in Remark (i) following their Corollary 5.4, and it concerns $G$-Galois algebras that are not fields. Corollary 5.4 shows: if $\mathrm{cd}_2(\Gamma_k)\le 2$, the $H^1$-condition holds, and $x_L^H=0$ for all $x\in H^2(H)$, then $L$ has an SDNB. When $\varphi$ is surjective — $L$ a $G$-Galois field extension — one has $H=G$, so the hypothesis reads: all degree-one and degree-two classes from $G$ vanish. For a general $G$-Galois algebra, however, $H=\varphi(\Gamma_k)$ is a proper subgroup of $G$, and the vanishing of the classes pulled back from $G$ is weaker than the vanishing of all classes from $H$. Whether this weaker vanishing already suffices is open; it is the subject of the present problem.

Problem Statement

Determine whether vanishing of the low-degree cohomology classes pulled back from $G$ forces the existence of a self-dual normal basis over fields of $2$-cohomological dimension at most $2$ — the last gap in the cohomological characterization of self-dual normal bases of $G$-Galois algebras.

Notation used throughout. For a field $k$, $\Gamma_k$ denotes its absolute Galois group and $\mathrm{cd}_2(\Gamma_k)$ its $2$-cohomological dimension, the smallest $d$ such that $H^i(\Gamma_k,C)=0$ for all $i>d$ and all finite $2$-primary $\Gamma_k$-modules $C$. For a finite group $G$, a $G$-Galois algebra over $k$ is a finite étale $k$-algebra $L$ with $\dim_k L=|G|$ on which $G$ acts by $k$-automorphisms with fixed subalgebra $k$; it corresponds to a continuous homomorphism $\varphi:\Gamma_k\to G$, well defined up to conjugacy. For $n\ge 1$, $\varphi^*:H^n(G,\mathbb{Z}/2\mathbb{Z})\to H^n(\Gamma_k,\mathbb{Z}/2\mathbb{Z})$ is the pullback in group cohomology with trivial $\mathbb{Z}/2\mathbb{Z}$-coefficients, and $x_L=\varphi^*(x)$. A self-dual normal basis (SDNB) of $L$ is a $k$-basis of the form $(gx)_{g\in G}$, for a single $x\in L$, that is orthonormal for the trace form $(x,y)\mapsto\mathrm{Tr}_{L/k}(xy)$. $G^2$ is the subgroup of $G$ generated by the squares of its elements, and $H=\varphi(\Gamma_k)$ denotes the image of $\varphi$.

Primary question (stated as an open question by Bayer-Fluckiger and Parimala, Remark (i) following Corollary 5.4 of their 2017 paper). Let $k$ be a field of characteristic $\ne 2$ with $\mathrm{cd}_2(\Gamma_k)\le 2$, let $G$ be a finite group, let $L$ be a $G$-Galois algebra over $k$, and let $\varphi:\Gamma_k\to G$ be a homomorphism corresponding to $L$. Assume:

(1) $x_L=\varphi^*(x)=0$ in $H^1(k,\mathbb{Z}/2\mathbb{Z})$ for every $x\in H^1(G,\mathbb{Z}/2\mathbb{Z})$ (the $H^1$-condition, i.e. $\varphi(\Gamma_k)\subseteq G^2$);

(2) $x_L=\varphi^*(x)=0$ in $H^2(k,\mathbb{Z}/2\mathbb{Z})$ for every $x\in H^2(G,\mathbb{Z}/2\mathbb{Z})$.

Does $L$ necessarily have a self-dual normal basis over $k$?

The boundary of the question is precisely mapped by the known results: the answer is yes over fields with $\mathrm{cd}_2(\Gamma_k)\le 1$ (then condition (1) alone suffices); yes when $\varphi$ is surjective, i.e. when $L$ is a $G$-Galois field extension, because then $H=\varphi(\Gamma_k)=G$ and conditions (1)-(2) are exactly the hypothesis of Corollary 5.4; no over fields of $2$-cohomological dimension $3$, where all positive-degree classes $x_L$ can vanish with no self-dual normal basis existing; and the converse implication is false in general (a self-dual normal basis can exist over $\mathbb{Q}$ with some degree-two class $x_L\ne 0$). By Theorem 5.3 of the 2017 paper the question is equivalent to: do conditions (1)-(2) imply $\mathrm{Res}_{E_A/k}(c_A(L))=0$ in $\mathrm{Br}_2(E_A)/\langle A\rangle$ for every orthogonal stable central simple factor $A$ of $k[G]_s$ and $\mathrm{Res}_{E_A/k}(d_A(L))=0$ in $\mathrm{Br}_2(E_A)$ for every unitary one? Here $k[G]_s$ is the semisimple quotient of the group algebra, the factors $A$ are taken stable under the canonical involution $\sigma$ with center $E_A$, $\mathrm{Br}_2(E_A)$ is the $2$-torsion of the Brauer group of $E_A$, and $c_A(L)$, $d_A(L)$ are the degree-two invariants of Bayer-Fluckiger-Parimala. The decisive case is that of $G$-Galois algebras that are not fields — induced algebras $\mathrm{Ind}_J^G M$ (induced from a $J$-Galois algebra $M$ for a subgroup $J\subseteq G$) and products of Galois extensions — where $H=\varphi(\Gamma_k)$ is a proper subgroup of $G$ and the degree-two classes pulled back from $G$ may vanish while the extension classes over $H$ that define the invariants $c_A(L)$ and $d_A(L)$ do not. The primary question is the goal of this problem.

Two further directions of the same source question are recorded for context only, without sharp completion boundaries of their own: (S1) beyond $\mathrm{cd}_2\le 2$, determine for which fields and groups the vanishing of all positive-degree classes $x_L$ is equivalent to the existence of a self-dual normal basis, and find the obstruction theory in $2$-cohomological dimension $\ge 3$ where all classes can vanish with no basis existing; (S2) give a terminating procedure (or prove impossibility in general) to compute the invariants $z_L$, $c_A(L)$ and $d_A(L)$ for concretely given $k$ and $L$, e.g. $k$ a global field and $L$ given by explicit polynomials, making the criterion of Theorem 5.3 algorithmic.

References

1. Eva Bayer-Fluckiger and Jean-Pierre Serre. Torsions quadratiques et bases normales autoduales. American Journal of Mathematics 116 (1994), no. 1, 1-64. DOI: 10.2307/2374981. https://doi.org/10.2307/2374981
2. Eva Bayer-Fluckiger and H. W. Lenstra, Jr. Forms in odd degree extensions and self-dual normal bases. American Journal of Mathematics 112 (1990), no. 3, 359-373. DOI: 10.2307/2374746. https://doi.org/10.2307/2374746
3. E. Bayer-Fluckiger and R. Parimala. Galois cohomology of the classical groups over fields of cohomological dimension at most 2. Inventiones Mathematicae 122 (1995), no. 1, 195-229. DOI: 10.1007/BF01231443. https://doi.org/10.1007/BF01231443
4. D.-S. Kang and Z. Reichstein. Trace forms of Galois field extensions in the presence of roots of unity. Journal für die reine und angewandte Mathematik 549 (2002), 79-89. DOI: 10.1515/crll.2002.067. https://doi.org/10.1515/crll.2002.067
5. Dong Seung Kang. Nonexistence of self-dual normal bases. Communications in Algebra 32 (2004), no. 1, 125-132. DOI: 10.1081/AGB-120027855. https://doi.org/10.1081/AGB-120027855
6. Eva Bayer-Fluckiger, Marina Monsurrò, R. Parimala and René Schoof. Trace forms of G-Galois algebras in virtual cohomological dimension 1 and 2. Pacific Journal of Mathematics 217 (2004), no. 1, 29-43. DOI: 10.2140/pjm.2004.217.29. https://doi.org/10.2140/pjm.2004.217.29
7. E. Bayer-Fluckiger, R. Parimala and J-P. Serre. Hasse principle for G-trace forms. Izvestiya: Mathematics 77 (2013), no. 3, 437-460. DOI: 10.1070/IM2013v077n03ABEH002643. https://doi.org/10.1070/IM2013v077n03ABEH002643
8. Vladimir Voevodsky. Reduced power operations in motivic cohomology. Publications Mathématiques de l'IHÉS 98 (2003), 1-57. DOI: 10.1007/s10240-003-0009-z. https://doi.org/10.1007/s10240-003-0009-z
9. Dmitri Orlov, Alexander Vishik and Vladimir Voevodsky. An exact sequence for $K^M_*/2$ with applications to quadratic forms. Annals of Mathematics (2) 165 (2007), no. 1, 1-13. DOI: 10.4007/annals.2007.165.1. https://doi.org/10.4007/annals.2007.165.1 (preprint arXiv:math/0101023, https://arxiv.org/abs/math/0101023, under the preprint title "An exact sequence for Milnor's K-theory with applications to quadratic forms")
10. Eva Bayer-Fluckiger and R. Parimala. Cohomological invariants for G-Galois algebras and self-dual normal bases. Documenta Mathematica 22 (2017), 1-24. DOI: 10.4171/dm/557. https://doi.org/10.4171/dm/557 (preprint arXiv:1608.04483, https://arxiv.org/abs/1608.04483)
11. François Arnault, Erik Jarl Pickett and Stéphane Vinatier. Construction of self-dual normal bases and their complexity. Finite Fields and Their Applications 18 (2012), no. 2, 458-472. DOI: 10.1016/j.ffa.2011.10.002. https://doi.org/10.1016/j.ffa.2011.10.002
12. Bruno Kahn. Galois descent for motivic theories. Theory and Applications of Categories 44 (2025), no. 20, 588-616. arXiv:2312.01825. https://arxiv.org/abs/2312.01825
13. Philippe Cassou-Noguès, Ted Chinburg, Baptiste Morin and Martin J. Taylor. On the trace form of Galois algebras. Bulletin of the London Mathematical Society 50 (2018), no. 5, 899-915. DOI: 10.1112/blms.12193. https://doi.org/10.1112/blms.12193 (preprint arXiv:1707.05049, https://arxiv.org/abs/1707.05049)