Pfister decomposition of the trace form of a degree-4 central simple algebra over fields without a primitive fourth root of unity

For a central simple algebra $A$ of degree 4 over a field $k$ of characteristic $\neq 2$ containing a square root $i$ of $-1$, Rost, Serre and Tignol proved that the trace form $q_A(x)=\operatorname{Trd}_A(x^2)$ decomposes uniquely in the Witt ring as $q_A=q_2+q_4$, where $q_2$ is the norm form of the quaternion algebra Brauer-equivalent to $A\otimes_k A$ and $q_4$ is a 4-fold Pfister form whose hyperbolicity characterizes cyclic (symbol) algebras; their proof uses the hypothesis $i\in k$ essentially. The characteristic-2 branch was settled in the same journal issue by Tignol, via the second trace form. It is open for which fields $k$ of characteristic $\neq 2$ in which $-1$ is not a square, and which degree-4 central simple $k$-algebras $A$, the same Pfister decomposition, with its algebraic interpretations, holds, and what replaces it when it fails.

Background

A central simple $k$-algebra (CSA) is a finite-dimensional algebra over a field $k$ whose center is exactly $k$ and which has no two-sided ideals other than $0$ and itself; by Wedderburn's theorem every CSA is isomorphic to a matrix algebra $M_r(D)$ over a division algebra $D$ with center $k$, and the degree of the algebra is $\sqrt{\dim_k A}$. Two CSAs are Brauer-equivalent when they have the same underlying division algebra up to isomorphism; Brauer classes form the Brauer group $\operatorname{Br}(k)$ under tensor product. A quaternion algebra $(a,b)_k$ is the 4-dimensional CSA generated by elements $u,v$ with $u^2=a$, $v^2=b$ and $vu=-uv$; quaternion algebras are the building blocks of the 2-torsion of $\operatorname{Br}(k)$, and a 2-torsion Brauer class of index at most 2 is the class of a unique quaternion algebra.

Throughout, $k$ has characteristic $\neq 2$, so a (nonsingular) quadratic form is a diagonal sum $\langle a_1,\dots,a_n\rangle$ whose dimension is $n$. The Witt ring $W(k)$ is the set of Witt classes of quadratic forms -- two forms are identified when they become isometric after adding hyperbolic planes $\langle 1,-1\rangle$ -- with addition given by orthogonal sum and multiplication by tensor product of forms. The fundamental ideal $I(k)\subset W(k)$ consists of the classes of even-dimensional forms, $I^n(k)$ is its $n$-th power, and an $n$-fold Pfister form is a tensor product

$$\langle\!\langle a_1,\dots,a_n\rangle\!\rangle=\langle 1,a_1\rangle\otimes\cdots\otimes\langle 1,a_n\rangle,$$

a form of dimension $2^n$ that is either anisotropic (does not represent $0$ nontrivially) or hyperbolic. The norm form of a quaternion algebra $(a,b)_k$ is the 2-fold Pfister form $\langle\!\langle -a,-b\rangle\!\rangle$. The Arason--Pfister Hauptsatz states that $I^n(k)$ contains no nonzero class represented by a form of dimension less than $2^n$; it is the basic dimension-counting principle of the Witt filtration. A form $\phi$ is said to be divisible by a form $\pi$ when $\phi\simeq\pi\otimes\psi$ for some form $\psi$.

The reduced trace $\operatorname{Trd}_A:A\to k$ is the ordinary matrix trace after base change to any splitting field of $A$ (a field over which $A$ becomes a matrix algebra), and it does not depend on the choice of splitting field. The trace form of a CSA $A$ of degree 4 is the 16-dimensional quadratic form

$$q_A(x)=\operatorname{Trd}_A(x^2),$$

a classical invariant of the Brauer class of $A$. For degree 4, $q_A$ has trivial discriminant (so $q_A\in I^2(k)$) and its Clifford invariant -- the second Stiefel--Whitney class $w_2(q_A)$, valued in the 2-torsion $\operatorname{Br}_2(k)$ of the Brauer group via the even Clifford algebra -- is the Brauer class of $A\otimes_k A$ when $-1$ is a square in $k$. For degree-4 algebras the class of $A\otimes_k A$ has index at most 2 (by a corestriction computation on a biquadratic maximal subfield, reproduced in the Rost--Serre--Tignol note), hence it is the class of a unique quaternion algebra $Q_A$, whose norm form is denoted $n_{Q_A}$.

A cyclic algebra of degree 4 over $k$ is a CSA generated by a cyclic Galois extension $L/k$ of degree 4 together with an element $v$ satisfying $vu=\sigma(u)v$ for a generator $\sigma$ of $\operatorname{Gal}(L/k)$. When $k$ contains a primitive fourth root of unity $i$ (an element with $i^2=-1$, that is, when $-1$ is a square in $k$), the cyclic algebras of degree 4 are exactly the symbol algebras, generated by $u,v$ with $u^4,v^4\in k$ and $vu=iuv$. In 2006, Rost, Serre and Tignol proved the following structure theorem. Suppose $\operatorname{char} k\neq 2$ and $i\in k$ with $i^2=-1$; then for every CSA $A$ of degree 4 over $k$ there exist a 2-fold Pfister form $q_2$ and a 4-fold Pfister form $q_4$ over $k$, uniquely determined, such that

$$q_A=q_2+q_4\quad\text{in }W(k),$$

and moreover: (1) $q_2$ is the norm form of the quaternion algebra Brauer-equivalent to $A\otimes_k A$; (2) $q_2$ and $q_4$ are divisible by the norm form of every quadratic extension of $k$ contained in $A$; (3) $q_2\cdot q_4=0$ in $W(k)$; and (4) $q_4$ is hyperbolic if and only if $A$ is a symbol, equivalently cyclic, algebra. As a corollary, the $j$-th exterior power $\lambda^j q_A$ (the induced form on the $j$-th exterior power of the underlying vector space) is $0$ for $j$ even and $q_A$ for $j$ odd, for $1\le j\le 15$ (with $\lambda^0 q_A=\lambda^{16}q_A=\langle 1\rangle$ and $\lambda^j q_A=0$ for $j>16$).

The proof rests on a classical theorem of Albert: every central division algebra of degree 4 over any field contains a maximal subfield that is Galois over the center with elementary abelian Galois group of order 4 (biquadratic in characteristic $\neq 2$). For such a subfield $K$, the orthogonal decomposition $A=K\oplus K_1\oplus K_2\oplus K_3$, where $K_1,K_2,K_3$ are the four-dimensional $k$-subspaces $K_j=\{x\in A\mid xy=\sigma_j(y)x\text{ for all }y\in K\}$ attached to the three nontrivial elements $\sigma_j$ of $\operatorname{Gal}(K/k)$, splits $q_A$ into four quaternary forms, and the decisive lemma shows that these quaternary forms compose (in the sense of the theory of compositions of quadratic forms) because the element $(1+i)xy+(1-i)yx$ always lies in a fixed component. The authors state in the note that the hypothesis $i\in k$ is used essentially in this step.

Through the norm-residue isomorphisms $I^n(k)/I^{n+1}(k)\simeq H^n(k,\mathbb{Z}/2)$, where $H^n(k,\mathbb{Z}/2)$ is $n$-th Galois cohomology with coefficients $\mathbb{Z}/2$, every $n$-fold Pfister form $\pi$ carries a cohomological class $e_n(\pi)\in H^n(k,\mathbb{Z}/2)$. The theorem therefore produces degree-2 and degree-4 cohomological invariants of degree-4 algebras, the latter detecting cyclicity. Since CSAs of degree 4 over $k$ are classified up to isomorphism by the nonabelian cohomology set $H^1(k,\operatorname{PGL}_4)$ for the projective linear group $\operatorname{PGL}_4$, these are cohomological invariants of the group $\operatorname{PGL}_4$; a theorem of Rost, proved in published form by Baek and Berhuy, shows they generate all mod-2 cohomological invariants of degree-4 algebras -- over base fields containing a square root of $-1$.

In characteristic 2 the first trace form degenerates ($\operatorname{Trd}(x^2)=\operatorname{Trd}(x)^2$), and the meaningful analogue is the second trace form, the second coefficient of the reduced characteristic polynomial of $x$, a quadratic form in the characteristic-2 sense, where a quadratic form and its polar bilinear form are distinct objects and nonsingular quadratic forms form a separate Witt group $W_q(k)$. In a companion note in the same journal issue, Tignol proved that over any field of characteristic 2 the second trace form of a degree-4 CSA is Witt-equivalent to $[1,1]+q_2+q_4$ in $W_q(k)$, where $[1,1]$ is the 2-dimensional form $x^2+xy+y^2$, $q_2$ is the norm form of the quaternion algebra Brauer-equivalent to $A\otimes_k A$, and $q_4=\langle 1,r_1\rangle\otimes\langle 1,r_2\rangle\otimes q_2$ is a 4-fold quadratic Pfister form, with $q_4=0$ again characterizing cyclic algebras. Both notes together thus leave exactly one case untouched: fields of characteristic $\neq 2$ in which $-1$ is not a square, that is, fields containing no primitive fourth root of unity.

Problem Statement

Let $k$ be a field of characteristic $\neq 2$ in which $-1$ is not a square (equivalently, $k$ contains no primitive fourth root of unity), and let $A$ be a central simple $k$-algebra of degree 4. Write $q_A(x)=\operatorname{Trd}_A(x^2)\in W(k)$ for its trace form, and $Q_A$ for the quaternion algebra over $k$ Brauer-equivalent to $A\otimes_k A$ (well defined because that class has index at most 2), with norm form $n_{Q_A}$.

(1) Existence of the Pfister decomposition. Do there exist a 2-fold Pfister form $q_2$ and a 4-fold Pfister form $q_4$ over $k$ such that $q_A=q_2+q_4$ in $W(k)$? Given existence, the pair $(q_2,q_4)$ is unique.

(2) Persistence of the algebraic interpretations. If the decomposition exists, do the identifications of the Rost--Serre--Tignol theorem persist over $k$: $q_2=n_{Q_A}$; $q_2\cdot q_4=0$ in $W(k)$; $q_2$ and $q_4$ divisible by the norm form of every quadratic extension of $k$ contained in $A$; and $q_4$ hyperbolic exactly when $A$ is cyclic over $k$? The criterion asked here is the one over $k$ itself; note that over the extension $k(i)$ cyclicity is the same as being a symbol algebra, so the formulations in terms of $A$ cyclic over $k$ and of $A_{k(i)}$ a symbol algebra are a priori different over $k$.

(3) Structure in case of failure. If the decomposition fails over some fields as in (1), determine what replaces it: identify the additional or different Witt-theoretic constituents occurring in $q_A$, and characterize how the Pfister constituents of the trace form of a degree-4 central simple algebra change when the base field contains no primitive fourth root of unity.

The common objective of (1)--(3) is a structure theorem for trace forms of degree-4 central simple algebras valid over all fields of characteristic $\neq 2$, recovering the Rost--Serre--Tignol theorem, including its corollary on the exterior powers $\lambda^j q_A$, when $-1$ is a square.

References

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