# Forms in $I^3_qF$ and their symbol length in $H_{2^3(F)}$

Adam Chapman, Ahmed Laghribi, Diksha Mukhija

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Adam Chapman, Ahmed Laghribi, Diksha Mukhija. Forms in $I^3_qF$ and their symbol length in $H_{2^3(F)}$. 2024. hal-04747198

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# FORMS IN $I_q^3 F$ AND THEIR SYMBOL LENGTH IN $H_2^3(F)$

ADAM CHAPMAN, AHMED LAGHRIBI, AND DIKSHA MUKHIJA

**ABSTRACT.** We obtain upper bounds on the symbol length of classes in $H_2^3(F)$ of forms in $I_q^3 F$ of any prescribed (even) dimension. The case of classes of forms in $I_q^3 F$ of dimension 14 will be studied in details.

## 1. INTRODUCTION

Let $F$ denote a field of characteristic 2. By [17], the cohomology groups $H_2^m(F)$ are isomorphic to $I_q^m F / I_q^{m+1} F$, which means the (classes of) quadratic $m$-fold Pfister forms generate the cohomology groups (see Section 2). Given a class in $H_2^m(F)$, we can define its symbol length to be the smallest number of forms in $GP_m F$ required in order to express this class.

In the special case of $m = 2$, the dimension $d$ of the quadratic form in $I_q^2 F$ provides the upper bound of $\frac{1}{2}d - 1$ for the symbol length of the class of this form in $H_2^2(F)$. This is an immediate result of the Clifford invariant, see for example [7].

The goal of this paper is to compute upper bounds for the symbol length in $H_2^3(F)$ of forms $\varphi \in I_q^3 F$ of prescribed dimension $2(n + 1)$. The general bound we provide is $\binom{2^{n+n+1}}{2}$ (Theorem 3.1). In the special case of $n = 5$, a known upper bound of 2 was given in [10], for such forms always decompose as $B \otimes [1, a]$, where $B$ is a bilinear Albert form. We elaborate a bit on forms of dimension 12 in $I_q^3 F$ and show that in general, they cannot be written as a multiple of a quadratic Albert form. We provide a bound of 3 for the symbol length of forms in $I_q^3 F$ of dimension 14 (Theorem 4.8) using a recent result of Hu, Laghribi and Sun showing that any anisotropic form in $I_q^3 F$ of dimension 14 contains an Albert quadratic form as a subform. We also produce an example showing that 3 is an optimal bound (Theorem 5.1). We also provide analogous results to the results of [25] in the case of symmetric bilinear forms in characteristic 2 (Theorem 6.10).

<footnote>Date: October 21, 2024.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary: 11E81; Secondary: 11E04; 19D45.</footnote>

<footnote>Key words and phrases. Quadratic Forms; Clifford Algebras; Arf Invariant; Symbol Length; Kato-Milne Cohomology; Fields of Characteristic Two.</footnote>
## 2. BACKGROUND ON QUADRATIC FORMS

Recall that any quadratic form $\varphi$ over $F$ can be written, up to isometry, as follows:

$$ (1) \qquad \varphi \cong [a_1, b_1] \perp \cdots \perp [a_r, b_r] \perp \langle c_1, \cdots, c_s \rangle, $$

where $[a, b]$ denotes the binary quadratic form $aX^2 + XY + bY^2$ for $a, b \in F$, and $\langle c_1, \cdots, c_s \rangle$ is the diagonal quadratic form $c_1X_1^2 + \cdots + c_sX_s^2$. The pair $(r, s)$ depends only on the isometry class of $\varphi$, we call it the type of $\varphi$. The form $\varphi$ is called nonsingular (resp. totally singular) if $s = 0$ (resp. $r = 0$).

The form $\varphi$ is isotropic if there exists a nonzero vector $v$ such that $\varphi(v) = 0$, otherwise it is called anisotropic.

For any form $\varphi$, there exists a unique anisotropic quadratic form, denoted by $\varphi_{an}$, such that

$$ \varphi \cong \varphi_{an} \perp [0, 0] \cdots \perp [0, 0] \perp \langle 0, \cdots, 0 \rangle. $$

The form $\varphi_{an}$ is called the anisotropic part of $\varphi$.

For $a, b, c \in F$ such that $b \neq 0$, let $[a; b; c]$ denote the nonsingular quadratic form given by the polynomial $aX^2 + bXY + cY^2$. It is easy to see that $[a; b; c] \cong [a, b^{-2}c]$.

Let $\varphi$ and $\psi$ be two quadratic forms of underlying vector spaces $V$ and $W$, respectively. We say that $\psi$ is dominated by $\varphi$, denoted by $\psi < \varphi$, if there exists an injective homomorphism $\sigma : W \to V$ such that $\varphi(\sigma(w)) = \psi(w)$ for all $w \in W$. We say that $\psi$ is a subform of $\varphi$, denoted by $\psi \subset \varphi$, if $\varphi \cong \psi \perp \psi'$ for a suitable quadratic form $\psi'$. It is clear that $\psi \subset \varphi$ implies that $\psi < \varphi$. In general, the converse in not true. However, if $\psi$ is nonsingular, then $\psi$ is dominated by $\varphi$ is equivalent to saying that $\psi$ is a subform of $\varphi$. We refer to [13, Section 3] for more details on the domination relation.

We say that two quadratic forms $\varphi$ and $\psi$ are Witt equivalent, denoted by $\varphi \sim \psi$, if $\varphi \perp m \times [0, 0] \cong \psi \perp n \times [0, 0]$ for some $m, n \in \mathbb{N}$.

For a nonsingular form $\varphi = [a_1, b_1] \perp \cdots \perp [a_r, b_r]$, its Arf invariant $\Delta(\varphi)$ is defined to be the class of $\sum_{i=1}^r a_i b_i$ in the the additive group $F/\varphi(F)$, where $\varphi(F) = \{x^2 + x \mid x \in F\}$. The Clifford algebra of $\varphi$ is denoted by $C(\varphi)$. Explicitly, we have $C(\varphi) \cong [b_1, a_1) \otimes_F \cdots \otimes_F [b_r, a_r)$, where for $a \in F^*$ and $b \in F$, we denote by $[b, a)$ the $F$-quaternion algebra generated by two elements $i$ and $j$ subject to the relations: $i^2 = a, j^2 + j = b$ and $iji^{-1} = j + 1$.

The term “bilinear form” means nondegenerate symmetric bilinear form of finite dimension. For $a_1, \cdots, a_n \in F^*$, we denote by $\langle a_1, \cdots, a_n \rangle_b$ the diagonal bilinear form $\sum_{i=1}^n a_i X_i Y_i$.

Given a bilinear form $B$ of underlying $F$-vector space $V$, we associate a totally singular quadratic form $\widetilde{B}$ having the same underlying vector space and defined by: $\widetilde{B}(v) = B(v, v)$ for all $v \in V$. The form $\widetilde{B}$ depends only on
the isometry class of $B$. However, a totally singular quadratic form could be associated to many bilinear forms.

An Albert quadratic form is a nonsingular quadratic form of dimension 6 and trivial Arf invariant. Similarly, an Albert bilinear form is a diagonal bilinear form of dimension 6 and trivial determinant.

Let $W_q(F)$ be the Witt group of nonsingular quadratic forms over $F$, and $W(F)$ the Witt ring of bilinear forms over $F$. It is well known that $W_q(F)$ is endowed with the $W(F)$-module structure in a natural way by tensor product [1].

For any integer $n \ge 1$, let $I^n F$ be the $n$-th power of the fundamental ideal $IF$ of $W(F)$ of even dimensional bilinear forms, and $I_q^{n+1} F = I^n F \otimes W_q(F)$ (we put $I^0 F = W(F)$ and thus $I_q^1 F = W_q(F)$). Each ideal $I^n F$ is additively generated by $n$-fold bilinear Pfister forms $\langle 1, a_1 \rangle_b \otimes \cdots \otimes \langle 1, a_n \rangle_b$ that we denote by $\langle\langle a_1, \cdots, a_n \rangle\rangle_b$. Hence, the subgroup $I_q^{n+1} F$ is additively generated by $(n+1)$-fold quadratic Pfister forms $\langle\langle a_1, \cdots, a_n \rangle\rangle_b \otimes [1, b]$ that we denote by $\langle\langle a_1, \cdots, a_n; b \rangle\rangle$. So we have natural filtrations of $W_q(F)$ and $W(F)$:

$$W_q F = I_q F \supseteq I_q^2 F \supseteq I_q^3 F \supseteq \dots$$

$$W(F) = I^0 F \supseteq IF \supseteq I^2 F \supseteq \dots$$

A fundamental result that we will use is the Hauptsatz of Arason-Pfister stating that any anisotropic form in $I_q^m F$ has dimension $\ge 2^m$ [11, Theorem 23.7].

Let $P_m F$ (resp. $GP_m F$) denote the set of isometry classes of $m$-fold quadratic Pfister forms (resp. the set of forms similar to $m$-fold quadratic Pfister forms). Similarly, for bilinear forms, we denote by $BP_m F$ the set of $m$-fold bilinear Pfister forms, and $GBP_m F$ the set of forms similar to $m$-fold bilinear Pfister forms.

A totally singular quadratic form $Q$ is called a quasi-Pfister form if there exists a bilinear Pfister form $B$ such that $Q \cong \widetilde{B}$.

A celebrated result of Kato connects the quotient $I_q^{n+1} F / I_q^{n+2} F$ to differential forms as follows: For $m \ge 0$, let $H_2^{m+1}(F)$ be the Kato-Milne cohomology group defined as the quotient

$$\Omega_F^m / (d\Omega_F^{m-1} + \varphi(\Omega_F^m)),$$

where $\Omega_F^m$ is the space of absolute $m$-differential forms, $d$ and $\varphi$ denote the differential and the Artin Schreier operators, respectively. We refer to [6] for the definitions.

In [17], Kato proved the following isomorphism:

$$e^m : \frac{I_q^{m+1} F / I_q^{m+2} F}{\langle\langle a_1, \dots, a_m; b \rangle\rangle} + I_q^{m+2} F \to \frac{H_2^{m+1}(F)}{b^{\frac{d a_1}{a_1} \wedge \dots \wedge \frac{d a_m}{a_m}}}$$
This isomorphism provides an alternative description of the cohomology
groups $H_2^{m+1}(F)$. Each element $\omega$ of $H_2^{m+1}(F)$ is a sum of symbols of the
shape $b \frac{da_1}{a_1} \wedge \cdots \wedge \frac{da_m}{a_m}$. It is natural to ask for the smallest number of such
symbols needed to express $\omega$. This number is called the symbol length
of $\omega$ and denoted by $\ell_m(\omega)$. Similarly, for any $\varphi \in I_q^{m+1}F$ anisotropic of
dimension $n$, we attach the invariant $\ell_{m,n}(\varphi)$ which is the smallest number
of forms in $GP_{m+1}F$ needed to write $\varphi$. Using the isomorphism $e^m$, one
can observe that $\ell_m(\omega) \le \ell_{m,n}(\varphi)$ when $\omega$ is the class of $\varphi$ in $H_2^{m+1}(F)$, i.e.,
$e^m(\varphi + I_q^{m+2}F) = \omega$. So in order to bound the symbol length for classes in
$H_2^{m+1}(F)$, we will work on the number $\ell_{m,n}(\varphi)$ for forms $\varphi \in I_q^{m+1}F$ of given
dimension $n$.

**Proposition 2.1.** Every form $\varphi \in I_q^2 F$ can be written similar to

$$
a_1[1, a_2] \perp a_2[1, a_3] \perp \cdots \perp a_n[1, a_{n+1}] \perp [1, a_2 + \cdots + a_{n+1}]
$$

for some choice of $a_1, \dots, a_{n+1} \in F$.

Proof. The form $\varphi = b_1[1, c_1] \perp \cdots \perp b_n[1, c_n] \perp [1, c_1 + \cdots + c_n]$ is
Witt equivalent to the sum of norm forms $N_1 \perp \cdots \perp N_n$ of the quaternion
algebras $[c_1, b_1)_{2,F}, \dots, [c_n, b_n)_{2,F}$. By [9], the algebras can be written as
$[a_2, a_1)_{2,F}, \dots, [a_{n+1}, a_n)_{2,F}$ for a certain choice of $a_1, \dots, a_{n+1} \in F$. There-
fore, $N_1 \perp \cdots \perp N_n$ is Witt equivalent to $a_1[1, a_2] \perp \cdots \perp a_n[1, a_{n+1}] \perp$
$[1, a_2 + \cdots + a_{n+1}]$. $\square$

3. UPPER BOUNDS FOR CLASSES IN $H_2^3(F)$

The proof of the following theorem follows almost the same line as [8, Theorem 3.6]. The main difference is that thanks to Proposition 2.1 that allows us to drop the assumption on the existence of the algebraic closure of $\mathbb{F}_2$ inside of $F$. We also avoid the black box of the Severi-Brauer variety and use Albert's results to give an explicit description of the field we descend to. The bound is also a little sharper.

**Theorem 3.1.** For every form $\varphi \in I_q^3 F$ anisotropic of dimension $2(n+1)$ over a field $F$ finitely generated over a perfect field $k$ of characteristic 2, we have $\ell_{2,2(n+1)}(\varphi) \le \binom{2^{n+n+1}}{2}$. In particular, the symbol length of its class in $H_2^3(F)$ is bounded by $\binom{2^{n+n+1}}{2}$.

Proof. By Proposition 2.1, one can choose $a_1, \dots, a_{n+1}$ such that $\varphi$ is similar
to $a_1[1, a_2] \perp a_2[1, a_3] \perp \dots \perp a_n[1, a_{n+1}] \perp [1, a_2 + \dots + a_{n+1}]$. Since the
class in $H_2^3(F)$ is invariant under multiplication of the form by a scalar, we
can assume the similarity above is an isometry. Let $K = k(a_1, \dots, a_{n+1})$.
Take $\pi = a_1[1, a_2] \perp a_2[1, a_3] \perp \dots \perp a_n[1, a_{n+1}] \perp [1, a_2 + \dots + a_{n+1}]$.
Then $C(\pi)$ is $[a_2, a_1)_{2,K} \otimes \dots \otimes [a_{n+1}, a_n)_{2,K}$. By [4, Chapter 7, Theorem
28], there exist a truncated Witt vector $\omega$ of length $n$ over $K$ and $\gamma \in K^\times$ such that $C(\pi) = [\omega, \gamma)_{2^n, K}$. Let $N : K_\omega \to K$ be the norm form of the underlying degree $2^n$ field extension. Since $\varphi = \pi_F$ and $C(\varphi)$ is trivial, we conclude the existence of $b_1, \dots, b_{2^n} \in F$ for which $N_F(b_1, \dots, b_{2^n}) = \gamma$. Set $L = K(b_1, \dots, b_{2^n})$. Then $\pi_L$ is in $I_q^3 L$. The symbol length of the class of $\pi_L$ in $H_2^3(L)$ is at most $\binom{2^n+n+1}{2}$ because the transcendence degree of $L$ over $k$ is at most $2^n+n+1$ and so its 2-rank is at most $2^n+n+1$ as well. Thus, the symbol length of the class of $\varphi$ in $H_2^3(F)$ is at most $\binom{2^n+n+1}{2}$ (see [8, Remark 3.1]). $\square$

### 4. FORMS IN $I_q^3 F$ OF DIMENSION UP TO 14

In addition to the Hauptsatz, we give another general result on forms in $I_q^n F$:

**Theorem 4.1.** Let $\varphi \in I_q^n F$ be an anisotropic quadratic form. If $\dim \varphi > 2^n$, then $\dim \varphi \ge 2^n + 2^{n-1}$.

*Proof.* See [21, Proposition 6.4]. $\square$

<figure><img src="image_2.png" /></figure>

We summarize the structure of forms in $I_q^3 F$ of dimension up to 12.

**Theorem 4.2.** Let $\varphi \in I_q^3 F$ be such that $\varphi$ is not hyperbolic. Then:

(1) $\dim \varphi_{an} \ge 8$.

(2) If $\dim \varphi = 8$, then $\varphi \in \mathrm{GP}_3(F)$, and thus $\ell_{2,8}(\varphi) = 1$.

(3) If $\dim \varphi = 10$, then $\varphi \sim \pi$ for some anisotropic $\pi \in \mathrm{GP}_3 F$. In particular, $\ell_{2,10}(\varphi) = 1$.

(4) If $\dim \varphi = 12$, then we have two cases:

(i) If $\varphi$ is isotropic, then there exists $\pi \in \mathrm{GP}_3 F$ such that $\varphi \sim \pi$. In particular, $\ell_{2,8}(\varphi) = 1$.

(ii) If $\varphi$ is anisotropic, then there exist $\gamma$ a bilinear Albert form and $a \in F^*$ such that $\varphi \cong \gamma \otimes [1, a]$. In particular, $\ell_{2,12}(\varphi) = 2$.

*Proof.* (1) is just the Hauptsatz. For (2) and (3) see [21, Proposition 6.4], and for (4) see [10]. $\square$

<figure><img src="image_3.png" /></figure>

The previous theorem makes use of the fact that any 12-dimensional form in $I_q^3 F$ is divisible by an Albert bilinear form, and thus it is split by a separable quadratic extension. However, the next proposition shows that such a form is not necessarily divisible by a quadratic Albert form.

**Proposition 4.3.** There exists a field $F$ of characteristic 2 and a form $\varphi \in I_q^3 F$ of dimension 12 that admit no Albert quadratic subform.

*Proof.* Consider $F_0 = \mathbb{F}_2^{sep}$, and $F = F_0(b_1, b_2, b_3, b_4, b_5, a)$ be the function field in 6 algebraically independent variables over $F_0$. Let $\varphi$ be the form
$\langle b_1, b_2, b_3, b_4, b_5, b_1b_2b_3b_4b_5 \rangle \otimes [1, a]$. This form is “incompressible” by [5], which means that there exists no field $E$ of transcendence degree less than 6 to which $\varphi$ descends, i.e., $\varphi \cong \rho_F$ for some form $\rho$ over $E$. Now, suppose that $\varphi$ admits an Albert quadratic subform $\gamma$. Then, there exists an Albert quadratic form $\gamma'$ such that $\varphi \cong \gamma \perp \gamma'$. It follows from [24] that $\gamma' \cong x\gamma$ for some $x \in F^*$. Hence, $\varphi = \langle 1, x \rangle_b \otimes \gamma$. By Proposition 2.1, $\gamma = c(a_1[1, a_2] \perp a_2[1, a_3] \perp [1, a_2 + a_3])$ for some $c, a_1, a_2, a_3 \in F$, and thus $\varphi$ descends to $F(x, c, a_1, a_2, a_3)$, a field of transcendence degree at most 5, a contradiction. $\square$

The following result due to Hu, Laghribi and Sun will play a crucial role to bound the symbol length of 14-dimensional quadratic forms in $I_q^3 F$.

**Theorem 4.4.** ([15, Theorem 7.10]) Let $\varphi \in I_q^3 F$ anisotropic of dimension 14. Then, $\varphi$ contains an Albert quadratic form as a subform.

**Corollary 4.5.** Let $\varphi \in I_q^3 F$ of dimension 14. Then, it is Witt equivalent to the sum of two forms in $I_q^3 F$ each of dimension 12.

*Proof.* By Theorem 4.4, $\varphi = \pi \perp \psi$ where $\pi$ is an Albert form and $\psi$ is of dimension 8. Let $x$ and $y$ be elements in $F^*$ represented by $\pi$ and $\psi$ respectively. Then, $\varphi$ is Witt equivalent to $\langle 1, \frac{y}{x} \rangle_b \otimes \pi \perp \tau$, where the two forms $\tau \in I_q^3 F$ and $\langle 1, \frac{y}{x} \rangle_b \otimes \pi \in I_q^3 F$ are of dimension at most 12. $\square$

**Corollary 4.6.** Any form in $I_q^3 F$ of dimension 14 is split by a separable bi-quadratic extension of $F$.

*Proof.* We combine Theorem 4.2(4) with Corollary 4.5. $\square$

One can ask whether there exist 14-dimensional forms in $I_q^3 F$ which has no separable quadratic splitting field. The answer is of course positive, and the argument is pretty easy.

**Example 4.7.** Let $\varphi$ be an anisotropic 14-dimensional form over $F$ which is Witt equivalent to the sum of two quadratic 3-fold Pfister forms $\psi$ and $\pi$ (such as the generic $\psi = \langle x, y, z \rangle$ and $\pi = \langle r, s, t \rangle$ over $\mathbb{F}_2(r, s, t, x, y, z)$), then it has no quadratic splitting field.

*Proof.* Suppose it has. Write the splitting field as $L = F[i : i^2 + i = a]$ for $a \in F \setminus \wp(F)$. Then, $\varphi = B \otimes [1, a]$ for some 7-dimension bilinear form $B$. However, this means that the Arf invariant of $\varphi$ is $\underbrace{a + \cdots + a}_{7 \text{ times}} = a$, but $\varphi$ is also in $I_q^2 F$, and so $a \in \wp(F)$, a contradiction. $\square$
A similar argument works also in characteristic not 2, and $\varphi$ cannot have an inseparable quadratic splitting field, because then $\varphi = \langle 1, b \rangle \otimes \rho$ for some $\rho \in IF$ where $\rho$ is of dimension 7, but all forms in $IF$ are of even dimension.

Combining Theorem 4.2(4) and Corollary 4.5, we get that $\ell_{3,14}(\varphi) \le 4$ for any $\varphi \in I_q^3 F$ anisotropic of dimension 14. The following theorem gives an improvement to this bound. Its analogue in charactersitic 2 is due to Hoffmann and Tignol [14, Prop. 2.3] and independently to Izhboldin and Karpenko [16, Prop. 17.2].

**Theorem 4.8.** Any anisotropic form $\varphi \in I_q^3 F$ of dimension 14 satisfies $\ell_{2,14}(\varphi) \le 3$. In particular, its class in $H_2^3(F)$ has length $\le 3$.

The proof of this theorem needs some preliminary results. We give two lemmas.

**Lemma 4.9.** Let $\varphi \in I_q^2 F$ of dimension 8 and $\tau \in P_2 F$ such that $\varphi_{F(\tau)} \in GP_3 F(\tau)$. Then, there exists $\pi \in GP_3 F$ such that $\varphi_{F(\tau)} \cong \pi_{F(\tau)}$.

*Proof.* If $\varphi \in GP_3 F$, there is nothing to prove. Suppose $\varphi \notin GP_3 F$, which means $\varphi \notin I_q^3 F$ (Theorem 4.2(2)), and thus $C(\varphi)$ is not split. The condition that $\varphi_{F(\tau)} \in GP_3 F(\tau)$ implies that $F(\tau)$ splits the algebra $C(\varphi)$. One knows that $\text{Ker}(\text{Br}_2(F) \to \text{Br}_2(F(\tau))) = \{0, [C(\tau)]\}$ (here $\text{Br}_2(F)$ denotes the 2-torsion of the Brauer group of $F$). Hence, $[C(\varphi)] = [C(\tau)]$, and thus $\varphi \perp \tau \in I_q^3 F$. Let $x \in D_F(\varphi)$. Then, $\varphi \perp x\tau \in I_q^3 F$. Since $\text{dim}(\varphi \perp x\tau)_{an} \le 10$, it follows from Theorem 4.2 that there exists $\pi \in GP_3 F$ such that $\varphi \perp x\tau \sim \pi$. Hence, $\varphi_{F(\tau)} \cong \pi_{F(\tau)}$. $\square$

The following lemma extends [14, Cor. 2.2] to characteristic 2, but here we propose a different proof.

**Lemma 4.10.** Let $\varphi \in I_q^2 F$ anisotropic of dimension $\le 10$ and $\tau \in P_2 F$ such that $\varphi \equiv \tau \in I_q^3 F$. Then, there exists $\pi \in GP_3 F$ and $x \in F^*$ such that $\varphi \sim \pi \perp x\tau$.

*Proof.* (1) Suppose that $\text{dim } \varphi = 10$. Since $\varphi_{F(\tau)} \in I_q^3 F(\tau)$, the form $\varphi_{F(\tau)}$ is isotropic, but it is not hyperbolic otherwise $\varphi$ would be isotropic. Hence, $\text{dim } (\varphi_{F(\tau)})_{an} = 8$. By the excellence of $F(\tau)/F$ [12, Cor. 5.7], there exists a form $\pi$ over $F$ such that $(\varphi_{F(\tau)})_{an} \cong \pi_{F(\tau)}$. By Lemma 4.9, we may suppose $\pi \in GP_3 F$. Since $(\varphi \perp \pi)_{F(\tau)}$ is hyperbolic, it follows that $\varphi \perp \pi \sim \rho \otimes \tau$ for $\rho$ an $F$-bilinear form. Further as the forms $\varphi \perp \pi$ and $\tau$ have the same Clifford invariant, we know $\rho$ is of dimension 1 or 3. So there exists $x \in F^*$ such that $\varphi \perp \pi \perp x\tau \in I_q^4 F$. Since $\text{dim } (\varphi \perp \pi \perp x\tau)_{an} \le 22$, there exists $\delta \in GP_4 F$ such that $\varphi \perp \pi \perp x\tau \sim \delta$ [21, Prop. 6.4]. Passing to $F(\pi)$, it is clear that $\delta_{F(\pi)}$ is isotropic and thus hyperbolic. Hence, we get $\delta \cong \langle r, s \rangle_b \otimes \pi$ for suitable $r, s \in F^*$. Now it is clear that $\varphi \perp x\tau \perp r s \pi \in I_q^5 F$. We get by the Hauptsatz that $\varphi \sim x\tau \perp r s \pi$.
(2) Suppose that $\dim \varphi = 8$. Since $\varphi_{F(\tau)} \in I_q^3 F(\tau)$ we get $\varphi_{F(\tau)} \in GP_3F(\tau)$. Hence, there exists $\pi \in GP_3F$ such that $\varphi_{F(\tau)} \cong \pi_{F(\tau)}$ (Lemma 4.9). We finish the proof as in the first case.

(3) Suppose that $\dim \varphi < 8$. Then, we get by the Hauptsatz that $\varphi_{F(\tau)} \sim 0$. Since $\varphi$ and $\tau$ have the same Clifford invariant, it follows that $\varphi \sim x\tau$ for some $x \in F^*$. In case we take for $\pi$ the hyperbolic form. $\square$

**Proof of Theorem 4.8.** Let $\varphi \in I_q^3 F$ be anisotropic of dimension 14. By [15, Theorem 7.10], there exists an Albert form $\alpha$ such that $\varphi \cong \alpha \perp \varphi'$ for $\varphi' \in I_q^2 F$ of dimension 8. Modulo a scalar, we may suppose that $\alpha \sim \tau_1 \perp \tau_2$ for $\tau_1, \tau_2 \in P_2 F$. Let $x \in D_F(\varphi')$. The form $\psi := (\varphi' \perp x\tau_1)_{an}$ has dimension $\le 10$. Moreover, we have

$$ \psi \sim \varphi' \perp x\tau_1 \sim \varphi \perp \tau_1 \perp \tau_2 \perp x\tau_1 \equiv \tau_2 \pmod{I_q^3 F}. $$

It follows from Lemma 4.10 that there exists $y \in F^*$ and $\pi \in GP_3F$ such that $\psi \sim \pi \perp y\tau_2$. Hence, we get $\varphi' \perp x\tau_1 \sim \pi \perp y\tau_2$. Consequently, $\varphi \sim \pi \perp \langle 1, x \rangle_b \otimes \tau_1 \perp \langle 1, y \rangle_b \otimes \tau_2$. $\square$

**5. AN OPTIMAL BOUND FOR 14-DIMENSIONAL FORMS IN $I_q^3 F$**

It is natural to ask if the bound 3 given in Theorem 4.8 for the symbol length of forms in $I_q^3 F$ of dimension 14 is optimal. The following theorem gives a positive answer to this question.

**Theorem 5.1.** There exist a field $K$ of characteristic 2 and a quadratic form $\varphi \in I_q^3 K$ of dimension 14 such that $\ell_{2,14}(\varphi) = 3$, i.e., $\varphi$ cannot be similar to the difference of two quadratic forms in $P_3 K$.

The proof of this theorem will be done in multiple steps and requires some details. We start with a definition:

**Definition 5.2.** (1) For any scalar $a \in F$ and field extension $K/F$, let $N_K(a)$ be the group $D_K([1, a])$ of scalars in $K^*$ represented by the quadratic form $[1, a]_K$.

(2) Let $E = F(\alpha)$ be a separable quadratic extension given by $\wp(\alpha) = r \in F \setminus \wp(F)$. For any two scalars $a, b \in F$, we denote by $N_1(a, b, r)$ the quotient group

$$ \frac{F^* \cap (N_E(a).N_E(b))}{(F^* \cap N_E(a)).(F^* \cap N_E(b))}. $$

The group $N_1(a, b, r)$ introduced above was used in [14] and will play a crucial role for the proof of Theorem 5.1, namely the construction of an example of a 14-dimensional form in $I_q^3 F$ whose symbol length in $H_2^3(F)$ is 3. We start by giving a property satisfied by this group.
**Lemma 5.3.** If $a \in F$ and $E = F(\alpha)$ a separable quadratic extension such that $\wp(\alpha) = r \in F \setminus \wp(F)$, then $F^* \cap N_E(a) = N_F(a).N_F(a+r)$.

*Proof.* (1) Let $d \in F^* \cap N_E(a)$. We have to prove that $d \in N_F(a).N_F(a+r)$. In fact, if $d \in N_F(a)$ then we are done. So suppose that $d \notin N_F(a)$. Then, $[1, a] \perp \langle d \rangle$ is anisotropic over $F$. Since this form becomes isotropic over $E$, there exists $t \in F^*$ such that $[1, a] \perp \langle d \rangle \cong t[1, r] \perp \langle d \rangle$. Using the completion lemma [13, Lem. 3.9], there exists $s \in F$ such that:

$$[1, a] \perp d[1, a + r] \cong t[1, r] \perp d[1, s].$$

Comparing Arf invariants on both sides, we get that $[1, s]$ is the hyperbolic plane. Hence, $[1, a] \perp d[1, a + r]$ is isotropic, which means that there exist $w \in D_F([1, a])$ and $w' \in D_F([1, a + r])$ such that $w = dw'$. Hence, $d = ww'^{-1} \in N_F(a).N_F(a+r)$.

(2) The inclusion $N_F(a).N_F(a+r) \subset F^* \cap N_E(a)$ is due to the isometry $[1, a+r]_E \cong [1, a]_E$. $\square$

**Theorem 5.4.** There exists a field $F$ of characteristic 2, and scalars $a, b \in F$ and $r \in F \setminus \wp(F)$ such that $N_1(a, b, r) \neq 1$.

*Proof.* Let $\mathbb{F}_2(x, y)$ be the rational function field in two variables $x$ and $y$ over the field $\mathbb{F}_2$ with two elements, and let $a = y, b = yx^4, c = y(1 + x^2 + x^4)$, and $E = F(\alpha)$ such that $\wp(\alpha) = c$. We have the following:

(1) $[1, a]_E$ represents $e := \alpha(x + x^2)$ by taking the vector $(\alpha, 1 + x + x^2)$.

(2) $[1, b]_E$ represents $f := \alpha(x^{-1} + x^{-2})$ by taking the vector $(\alpha, 1 + x^{-1} + x^{-2})$.

Let us take $z = x^{-1}$. So we can write $e = \alpha z^{-2}(1 + z)$ and $f = \alpha z(1 + z)$. Hence, $d := e^{-1}f = z^3 \in F^* \cap N_E(a).N_E(b)$.

**Claim.** We have $d \notin (F^* \cap N_E(a)).(F^* \cap N_E(b))$.

To prove the claim, it is sufficient to check that $d \notin N_F(a).N_F(a+c).N_F(b).N_F(b+c)$ (Lemma 5.3). Since $d = z^3$, it suffices to prove that the elements represented by the groups $N_F(a), N_F(a+c), N_F(b)$ and $N_F(b+c)$ have even $z$-adic valuation. In fact:

(a) Let $A \in N_F(a)$. Then, there exist $u, v, w \in \mathbb{F}_2(y)[z]$ and $\epsilon \in \mathbb{N}$ even such that $A = w^{-2}z^\epsilon(u^2 + uv + yv^2)$ and $u$ and $v$ are not simultaneously divisible by $z$. Since the form $[1, y]$ is anisotropic over $\mathbb{F}_2(y)$, and $u$ and $v$ are not simultaneously divisible by $z$, it follows that $u^2 + uv + yv^2$ is a unit. Hence, the valuation of $A$ is even.

(b) Let $B \in N_F(a+c)$. We have $a+c = y(x^2 + x^4)$. As in the case (a), there exist $u, v, w \in \mathbb{F}_2(y)[z]$ and $\epsilon \in \mathbb{N}$ even such that $B = w^{-2}z^\epsilon(u^2 + uv + y(x^2 + x^4)v^2)$ and $u$ and $v$ are not simultaneously divisible by $z$. We multiply and divide by $z^4$, we get: $B = w^{-2}z^{\epsilon-4}(z^4u^2 + z^4uv + y(z^2+1)v^2)$. It is easy to show that the $z$-adic valuation of $z^4u^2 + uvz^4 + y(z^2+1)v^2$ is even (using
the fact that ⟨1, y⟩ is anisotropic over $\mathbb{F}_2(y)$ and discussing on $v$ if it is a unit or not). Hence, the $z$-adic valuation of $B$ is even.

(c) Similarly, we prove that each element represented by $N_F(b)$ and $N_F(b + c)$ has an even $z$-adic valuation. $\square$

Let $K$ be a field which is complete with respect to a discrete valuation. Let $A$ be its ring, $\pi$ a uniformizer and $\kappa$ the residue field. Let $\varphi$ be an anisotropic quadratic form over $K$ whose underlying vector space is $V$. For any $i \in \{0, 1, 2\}$, the set $V_i = \{v \in V \mid \varphi(v) \in \pi^i A\}$ is an $A$-module. The first and second residue forms of $\varphi$, denoted by $\overline{\varphi_0}$ and $\overline{\varphi_1}$ respectively, are defined as follows:

$$ \overline{\varphi_i}: \begin{array}{ccc} V_i/V_{i+1} & \to & \kappa \\ v + V_{i+1} & \mapsto & \pi^{-i}\varphi(v) \end{array} $$

We refer to [23] for further details concerning the residue forms and their computation in general. Here, we give an explicit computation of residue forms of a 2-fold quadratic Pfister form.

**Lemma 5.5.** Let $K = F((t))$ be the field of Laurent series in the variable $t$ over $F$. Let $\sigma \in \text{GP}_2 K$ and $\overline{\sigma_0}, \overline{\sigma_1}$ its first and second residue forms with respect to the $t$-adic valuation of $K$. Then, we have the following statements:

(1) $\dim \overline{\sigma_0} + \dim \overline{\sigma_1} = 4$.

(2) $(\dim \overline{\sigma_0}, \dim \overline{\sigma_1}) \in \{(4, 0), (0, 4), (2, 2)\}$.

(3) If $\dim \overline{\sigma_i} = 4$ for $i = 0$ or $1$, then $\overline{\sigma_i} \in \text{GP}_2(F)$ or $\overline{\sigma_i}$ is similar to a quasi-Pfister form.

(4) If $\dim \overline{\sigma_0} = \dim \overline{\sigma_1} = 2$, then $\overline{\sigma_0}$ is similar to $\overline{\sigma_1}$.

*Proof.* Let $A$ be the ring of the $t$-adic valuation $v$ of $K$. We attribute (1) to [23, Theorem 1]. To deduce (2), (3) and (4) we follow the same steps given in [23, Page 346].

Let $a, b, c, u \in A$ be such that $\sigma \cong [a; b; c] \perp u[a; b; c]$. Moreover, we choose such a decomposition so that $v(b)$ is minimal. Moreover, we may suppose $v(a), v(c) \in \{0, 1\}$ and $u$ is squarefree. Let us denote by $\delta$ (resp. $\eta$) the form $[a; b; c]$ (resp. the form $u[a; b; c]$). Now we distinguish between different cases:

(i) Suppose $v(b) = 0$. By the Schwarz inequality [23, Eq. (4)], we get $v(a) = v(c) = 0$. We have $\overline{\delta_0} = [\overline{a}; \overline{b}; \overline{c}]$ and $\overline{\eta_0} = \overline{u}[\overline{a}; \overline{b}; \overline{c}]$. As proved in [23, Case (i)], we have $D_F(\overline{\delta_0}) \cap D_F(\overline{\eta_0}) = \{0\}$. Hence,

$$ (\overline{\sigma_0}, \overline{\sigma_1}) = \begin{cases} ([\overline{a}; \overline{b}; \overline{c}] \perp \overline{u}[\overline{a}; \overline{b}; \overline{c}], 0) & \text{if } v(u) = 0, \\ ([\overline{a}; \overline{b}; \overline{c}], \overline{t^{-1}u}[\overline{a}; \overline{b}; \overline{c}]) & \text{if not.} \end{cases} $$
(ii) Suppose $v(b) \ge 1$ and $v(a) = v(c) = 1$. As in case (1), we have $D_F([t^{-1}a; t^{-1}b; t^{-1}c]) \cap D_F(\bar{\eta}_1) = \{0\}$. Hence, we get:

$$ (\bar{\sigma}_0, \bar{\sigma}_1) = \begin{cases} \left(0, [\overline{t^{-1}a}; \overline{t^{-1}b}; \overline{t^{-1}c}] \perp \overline{u}[\overline{t^{-1}a}; \overline{t^{-1}b}; \overline{t^{-1}c}]\right) & \text{if } v(u) = 0, \\ \left(\overline{t^{-1}u}[\overline{t^{-1}a}; \overline{t^{-1}b}; \overline{t^{-1}c}], [\overline{t^{-1}a}; \overline{t^{-1}b}; \overline{t^{-1}c}]\right) & \text{if not.} \end{cases} $$

(iii) Suppose $v(b) > 1$ and $v(a) = v(c) = 1$. Hence, $\bar{\delta}_0 = 0$ and $\bar{\delta}_1 = \langle t^{-1}a, t^{-1}c \rangle$. As in [23, Case (iii)], we get:

$$ (\bar{\sigma}_0, \bar{\sigma}_1) = \begin{cases} \left(\langle \overline{t^{-1}a}, \overline{t^{-1}c} \rangle \perp \overline{u} \langle \overline{t^{-1}a}, \overline{t^{-1}c} \rangle\right) & \text{if } v(u) = 0, \\ \left(\overline{t^{-1}u} \langle \overline{t^{-1}a}, \overline{t^{-1}c} \rangle, \langle \overline{t^{-1}a}, \overline{t^{-1}c} \rangle\right) & \text{if not.} \end{cases} $$

(iv) Suppose $v(b) \ge 1$, $v(a) = 0$, $v(c) = 1$ and $D_F(\langle \bar{a} \rangle) \cap D_F(\bar{\eta}_0) = \{0\}$. We have $\bar{\delta}_0 = \langle \bar{a} \rangle$ and $\bar{\delta}_1 = \langle t^{-1}c \rangle$. Furthermore, as proved in [23, Case (iv)], we have $D_F(\bar{\delta}_1) \cap D_F(\bar{\eta}_1) = \{0\}$. Hence, we get:

$$ (\bar{\sigma}_0, \bar{\sigma}_1) = \begin{cases} \langle \bar{a}, \overline{ua} \rangle, \langle \overline{t^{-1}c}, \overline{ut^{-1}c} \rangle & \text{if } v(u) = 0, \\ \langle \bar{a}, \overline{t^{-2}uc} \rangle, \langle \overline{t^{-1}c}, \overline{t^{-1}ua} \rangle & \text{if not.} \end{cases} $$

(v) Suppose $v(b) \ge 1$, $v(a) = 0$, $v(c) = 1$ and $D_F(\langle \bar{a} \rangle) \cap D_F(\bar{\eta}_0) \ne \{0\}$. As in [23, Case (v)], there exists $d, u \in A$ such that $\varphi \cong [td; b; c] \perp u'[td; b; c]$ and $vtd) = 1$. Hence, we can apply the case (ii) or (iii).

(vi) Suppose $v(b) \ge 1$, $v(a) = v(c) = 0$ and $D_F(\langle \bar{a}, \bar{c} \rangle) \cap D_F(\bar{\eta}_0) = \{0\}$. As explained in [23, Case (vi)], we may suppose $\langle \bar{a}, \bar{c} \rangle$ anisotrope, and we get:

$$ (\bar{\sigma}_0, \bar{\sigma}_1) = \begin{cases} \langle \bar{a}, \bar{c} \rangle \perp \overline{u} \langle \bar{a}, \bar{c} \rangle, 0 & \text{if } v(u) = 0, \\ \langle \bar{a}, \bar{c} \rangle, \overline{t^{-1}u} \langle \bar{a}, \bar{c} \rangle & \text{if not.} \end{cases} $$

(vii) Suppose $v(b) \ge 1$, $v(a) = v(c) = 0$ and $D_F(\langle \bar{a}, \bar{c} \rangle) \cap D_F(\bar{\eta}_0) \ne \{0\}$. As explained in [23, Case (vii)], we may find another decomposition of $\varphi$ that reduces to the case (iv) or (v). $\square$

**Lemma 5.6.** Let $K = F((t))$, and $\psi_0, \psi_1$ two nonsingular quadratic forms over $F$ of dimension 4. Let $\sigma$ be a form similar to a quasi-Pfister form of dimension 4 over $K$ such that $\sigma < \psi_0 \perp t\psi_1$. Let $\bar{\sigma}_0, \bar{\sigma}_1$ be the first and the second residue forms of $\sigma$ with respect to the $t$-adic valuation of $K$. Then, $\bar{\sigma}_i < \psi_i$ for $i = 0, 1$, and the forms $\bar{\sigma}_0$ and $\bar{\sigma}_1$ are both of dimension 2 and similar.

*Proof.* Let $\varphi = \psi_0 \perp t\psi_1$. The first and the second residue forms of $\varphi$ are $\psi_0$ and $\psi_1$, respectively. Hence, $\bar{\sigma}_i < \psi_i$ for $i = 0, 1$. We know that $\bar{\sigma}_0$ and $\bar{\sigma}_1$ are totally singular as $\sigma$ is also totally singular. Without loss of generality, we may reduce to the case where $F$ has a finite 2-basis, and thus $K = F((t))$ is defectless, meaning that $[K : K^2] = 2[F : F^2]$. It follows from [23, Th. 1] that $\dim \bar{\sigma}_0 + \dim \bar{\sigma}_1 = 4$. Since $\dim \sigma = 4$ and the forms
$\psi_0$ and $\psi_1$ are nonsingular quadratic forms of dimension 4, it follows that $\dim \overline{\sigma_0} = \dim \overline{\sigma_1} = 2$.

It remains to prove that $\overline{\sigma_0}$ and $\overline{\sigma_1}$ are similar. In fact, let $V$ be the underlying $K$-vector space of $\sigma$, and $V_i = \{v \in V \mid \sigma(v) \in t^i A\}$ for $0 \le i \le 2$, where $A$ is the ring of the $t$-adic valuation of $K$.

Let $v_1, v_2 \in V_0$ and $w_1, w_2 \in V_1$ be such that $\{v_1 + V_1, v_2 + V_1\}$ is an $F$-basis of $V_0/V_1$, and $\{w_1 + V_2, w_2 + V_2\}$ is an $F$-basis of $V_1/V_2$. We take $\sigma(v_i) = \alpha_i$ and $\sigma(w_i) = \beta_i$ for $i = 1, 2$. Hence, $\overline{\sigma_0} \cong \langle \overline{\alpha_1}, \overline{\alpha_2} \rangle$, and $\overline{\sigma_1} \cong \langle t^{-1}\beta_1, t^{-1}\beta_2 \rangle$. Since $\{v_1, v_2, w_1, w_2\}$ is an $F$-basis of $V$, we get $\sigma \cong \langle \alpha_1, \alpha_2, \beta_1, \beta_2 \rangle$. Using the roundness of quasi-Pfister forms, we deduce that $\sigma \cong \langle \alpha_1, \alpha_2, \beta_1, \alpha_1\alpha_2\beta_1 \rangle$. Hence, we deduce that $\overline{\sigma_1} \cong t^{-1}\alpha_1\beta_1 \langle \overline{\alpha_1}, \overline{\alpha_2} \rangle$, as desired. $\square$

The following lemma extends [14, Lemma 3.3] to characteristic 2. It gives a behavior property over quadratic extensions of 8-dimensional forms in $I_q^2 F$ that have a Clifford algebra of index 4. We don't exclude the case of inseparable quadratic extensions, which is why we include the forms similar to quasi-Pfister forms of dimension 4 in statement (1) of the following lemma. This is the unique difference comparing to the characteristic not 2 version.

**Lemma 5.7.** Let $\varphi \in I_q^2 F$ of dimension 8 such that $C(\varphi)$ has index 4. Let $A$ be a biquaternion algebra such that $[C(\varphi)] = [A]$. Let $\alpha$ be an Albert form associated to $A$. Then, the following statements are equivalent:

(1) $\varphi$ dominates a form of dimension 4 which is either in $GP_2(F)$ or is similar to a quasi-Pfister form.

(2) There exists $L/F$ a quadratic extension such that $\varphi_L$ and $\alpha_L$ are both isotropic.

(3) There exists $L/F$ a quadratic extension such that $\varphi_L$ is isotropic and $A_L$ is not a division algebra.

*Proof.* The equivalence (2) $\iff$ (3) follows from Albert's result.

(1) $\implies$ (2) Suppose that $\varphi$ dominates a form $\sigma$ of dimension 4 such that $\sigma \in GP_2F$ or $\sigma$ is similar to a quasi-Pfister form. Let $L/F$ be a quadratic extension such that $\sigma_L$ is isotropic. Then, $i_W(\sigma_L) = 2$ or $i_d(\sigma_L) = 2$ as $\sigma \in GP_2F$ or $\sigma$ is similar to a quasi-Pfister form. Hence, $i_W(\varphi_L) \ge 2$. Consequently, $\varphi_L \sim \tau$ for a suitable form $\tau \in GP_2(L)$. Since $[C(\varphi_L)] = [C(\tau)] = [C(\alpha_L)]$, it follows that $\dim C(\alpha_L) \le 2$, which implies that $\alpha_L$ is isotropic. Hence, $\varphi_L$ and $\alpha_L$ are both isotropic.

(2) $\implies$ (1) Let $L/F$ be a quadratic extension such that $\varphi_L$ and $\alpha_L$ are isotropic.

(a) Suppose that $L = F(u)$ such that $\wp(u) = a \in F \setminus \wp(F)$. Since $\varphi_L$ is isotropic, we get $\wp \cong x[1, a] \perp \varphi'$ for suitable $x \in F^*$ and $\varphi'$ of dimension 6.
over $L$. Passing to $L$, the form $\varphi'_L$ is an Albert form and $\varphi_L \sim \varphi'_L$. We have $[C(\varphi_L)] = [C(\varphi'_L)] = [C(\alpha_L)]$. Since $\alpha_L$ is isotropic, it follows that $\varphi'_L$ is isotropic as well. Hence, $\varphi' \cong y[1, a] \perp \varphi''$ for some $y \in F^*$ and $\varphi''$ a form of dimension 4. Consequently, $\varphi \cong \langle x, y \rangle_b \otimes [1, a] \perp \varphi''$. So we are done in this case.

(b) Suppose that $L = F(\sqrt{d})$ such that $d \in F \setminus F^2$. Since $\varphi_L$ is isotropic, we get $\varphi \cong x[1, k] \perp xd[1, l] \perp \varphi'$ for suitable $x, k, l \in F^*$ and $\varphi'$ a form of dimension 4. Passing to $L$, we get $\varphi_L \sim (x[1, k+l] \perp \varphi')_L$. Using the fact that $\alpha_L$ is isotropic, we conclude as in the case (a) that the Albert form $(x[1, k+l] \perp \varphi')_L$ is isotropic. Hence, we get

$$x[1, k+l] \perp \varphi' \cong y[1, m] \perp yd[1, n] \perp \varphi''$$

for some $y, m, n \in F^*$ and $\varphi''$ a form of dimension 2. Consequently, we have

$$ (2) \quad \begin{aligned} \varphi &\sim x[1, k] \perp xd[1, l] \perp x[1, k+l] \perp y[1, m] \perp yd[1, n] \perp \varphi'' \\ &\sim x[1, l] \perp xd[1, l] \perp y[1, m] \perp yd[1, n] \perp \varphi'' \\ &\sim x\pi \perp \gamma, \end{aligned} $$

where $\pi = \langle\langle d, xy, l \rangle\rangle \in P_3(F)$ and $\gamma = y[1, l+m] \perp yd[1, l+n] \perp \varphi''$ is an Albert form. Note that $\pi$ is anisotropic, otherwise $\varphi$ would be isotropic by equation (2). We discuss on $i_W(\gamma)$:

(i) Suppose $i_W(\gamma) = 2$ or 3, then $\varphi \cong x\pi$, and we are done.

(ii) Suppose $i_W(\gamma) = 1$. Then $\gamma \sim \tau$ for a suitable $\tau \in GP_2(F)$. In this case, $i_W(x\pi \perp \tau) = 2$. Hence, $x\pi$ and $\tau$ dominated a common form $\delta$ of dimension 2 [13, Prop. 3.11]. According as $\delta$ is singular or not, we conclude that $\varphi$ contains a Pfister neighbor of type (2, 1) or (3, 0), which implies that $\varphi$ contains a subform in $GP_2(F)$ using the classification of Pfister neighbors of dimensions 5 and 6 given in [18, Prop. 3.2].

(iii) Suppose $i_W(\gamma) = 0$. Since $i_W(x\pi \perp \gamma) = 3$, it follows that $x\pi$ and $\gamma$ dominate a common form $\eta$ of dimension 3 [13, Prop. 3.11].

– Suppose that $\eta$ is of type (1, 1). Then, $\varphi$ dominates a Pfister neighbor of type (2, 1). Again we deduce from [18, Prop. 3.2] that $\varphi$ contains a subform in $GP_2F$.

– Suppose that $\eta$ is of type (0, 3). Let us write $\eta = \langle u, v, w \rangle$. The form $\eta' := \eta \perp \langle uvw \rangle$ is anisotropic. Since $\pi$ is isotropic over $F(\eta)$, it becomes isotropic over $F(\eta')$. Using the roundness of $\pi$ and the fact that $\eta < x\pi$, we conclude that $\eta' < x\pi$. Hence, $\eta' < \varphi$, and we are done. $\square$

We recall an elementary lemma due to Aravire and Baeza [2, Lem. 2.1]:

**Lemma 5.8.** Let $a, b \in F^*$ be such that $a + b \neq 0$. Then, we have:

$$\langle 1, a+b \rangle_b \otimes [1, u] \sim \langle 1, a \rangle_b \otimes \left[ 1, \frac{au}{a+b} \right] \perp \langle 1, b \rangle_b \otimes \left[ 1, \frac{bu}{a+b} \right].$$
We deduce the following corollary.

**Corollary 5.9.** Let $a, b \in F^*$ be such that $a^2 + b \neq 0$. Then, we have

$$ \langle 1 \rangle \perp (a^2 + b)[1, u] \sim \langle 1 \rangle \perp b \left[ 1, \frac{bu}{a^2 + b} \right]. $$

*Proof.* Using Lemma 5.8, we get $[1, u] \perp (a^2+b)[1, u] \sim [1, u] \perp b[1, \frac{bu}{a^2+b}]$. Adding the form $\langle 1 \rangle$ on both sides and using the fact that $\langle 1 \rangle \perp [1, u] \sim \langle 1 \rangle$, we get the conclusion. □

We start by giving some properties attached to $F$ which were studied by Hoffmann and Tignol in characteristic not 2 [14]. We adapt the formulation of some of them to characteristic 2.

*Property D(2):* For every quadratic separable extension $E/F$, and all non-singular $F$-quadratic forms $\varphi$ and $\psi$ of dimension 2, if $D_E(\varphi) \cap D_E(\psi) \neq \emptyset$ then $D_E(\varphi) \cap D_E(\psi) \cap F^* \neq \emptyset$.

*Property CS:* For every separable quadratic extension $E/F$, if two quater-nion algebras over $F$ have a common slot over $E$ (separable or inseparable), then they have a common slot in $F$.

*Property D(4):* For all 4-dimensional nonsingular $F$-quadratic forms $\varphi_1$ and $\varphi_2$ such that $\Delta(\varphi_1) = \Delta(\varphi_2)$, if there exists a separable quadratic extension $E/F$ such that $(\varphi_1)_E \cong (\varphi_2)_E$, then there exists a binary form $\beta$ which is dominated up to a scalar by both $\varphi_1$ and $\varphi_2$.

*Property D(8):* Every 8-dimensional form $\varphi \in I_q^2 F$ whose Clifford algebra is of index 4 dominates a form of dimension 4 which is either in $GP_2 F$ or is similar to a quasi-Pfister form.

*Property D(14):* Every 14-dimensional form $\varphi \in I_q^3 F$ is similar to the difference of two forms in $P_3 F$.

There are strong connections between these properties as it was proved by Hoffmann and Tignol in characteristic not 2 [14]. Here we explain how we can get a counter-example to some property knowing a counter-example to another one. More precisely, this phenomenon proceeds from property $D(2)$ until property $D(14)$. This is our aim in the following proposition which extends [14, Corollary 6.2]:

**Proposition 5.10.** Let $F$ be a field of characteristic 2 for which there exist $a, b, c$ satisfying $N_1(a, b, c) \neq 1$. Let $E = F(\varphi^{-1}(c))$ and $d \in F^* \cap D_E([1, a])$. $D_E([1, b])$ such that $d \notin (F^* \cap D_E([1, a])) \cdot (F^* \cap D_E([1, b]))$. Let $t_1, t_2, t_3$ be independent variables over $F$, $F_1 = F((t_1))$, $F_2 = F_1((t_2))$, $F_3 = F_2((t_3))$ and $E_i = F_i(\varphi^{-1}(c))$ for $i = 1, 2, 3$. We have:

(1) The forms $[1, a]_E$ and $d[1, b]_E$ represent a common scalar over $E$, but
there does not exist a scalar in $F^*$ represented by the forms $[1, a]_E$ and $d[1, b]_E$.

(2) The two quaternion algebra $[a, t_1]_{E_1}$ and $[b, dt_1]_{E_1}$ have a common slot over $E_1$, but they do not have a common slot in $F_1^*$.

(3) Let $\psi_1 := [1, a + c] \perp t_1[1, a]$ and $\psi_2 = [1, b + c] \perp dt_1[1, b]$. Then, there exist $u, v \in F_1^*$ such that over $L := F_1(\wp^{-1}(u))$ we have $(\psi_1)_L \cong v(\psi_2)_L$, but there does not exist a binary form over $F_1$ which is similar to a form dominated by $\psi_1$ and $\psi_2$.

(4) The Clifford invariant of the form $\psi := \psi_1 \perp t_2\psi_2 \in I_q^2 F_2$ is represented by a biquaternion $F_2$-algebra $A$, but $\psi$ does not dominate a form of dimension 4 in $GP_2F_2$ or similar to a quasi-Pfister form.

(5) For $A$ as in statement (4), let $\alpha$ be an Albert form associated to $A$, and let $\wp = \psi \perp t_3\alpha$. Then, $\wp \in I_q^3 F_3$, $\dim \wp = 14$ and $\wp$ is not similar to the difference of two forms in $P_3F_3$.

*Proof.* (1) Let $r \in D_E([1, a])$ and $s \in D_E([1, b])$ be such that $d = rs$. Then, $r = ds^{-1} \in D_E([1, a]) \cap D_E(d[1, b])$. Suppose that there exists $\alpha \in F^* \cap D_E([1, a]) \cap D_E(d[1, b])$. Hence, $\alpha^{-1}d^2 \in F^* \cap D_E([1, a])$ and $\alpha d^{-1} \in F^* \cap D_E([1, b])$. Consequently, $d = \alpha^{-1}d^2.(\alpha d^{-1}) \in (F^* \cap D_E([1, a]).(F^* \cap D_E([1, b]))$, a contradiction.

(2) Since $D_E([1, a]) \cap D_E(d[1, b]) \neq \emptyset$, the quaternions algebras $[a, t_1]_{E_1}$ and $[b, dt_1]_{E_1}$ have a common inseparable slot over $E_1$. Suppose they have a common slot in $F_1$, separable or inseparable. Namely, they share a common quadratic field extension of their center $E_1$, that descends to a quadratic extension of $F_1$ (i.e., it is $L \otimes_{F_1} E_1$ for some $L/F$ of degree 2). Since the residue algebras are disjoint, $L \otimes_{F_1} E_1/E_1$ is ramified, which means $L/F_1$ is ramified as well. Suppose $L/F_1$ is separable. Then

$$a + y + y^2 + t_1(z^2 + zw + aw^2) = b + x + x^2 + dt_1(r^2 + rs + s^2b)$$

has a solution $x_0, y_0, z_0, w_0, q_0, r_0, s_0$ in $E_1$ such that $a+y_0+t_1(z_0^2+z_0w_0+aw_0^2)$ is in $F_1$. Since $L/F$ is ramified, $y_0$ and $x_0$ can be chosen so that the value of $a + y_0 + t_1(z_0^2 + z_0w_0 + aw_0^2)$ is odd. Therefore, in the residue field, we have $\overline{z_0^2} + \overline{z_0w_0} + \overline{aw_0^2} = d(\overline{r_0^2} + \overline{r_0s_0} + \overline{bs_0^2})$ and $\overline{z_0^2} + \overline{z_0w_0} + \overline{aw_0^2} \in F$. The latter contradicts the assumption that $d \notin (F^* \cap D_E([1, a]).(F^* \cap D_E([1, b]))$. The argument when $L/F_1$ is inseparable is similar, we just start with the equation

$$y^2 + t_1(z^2 + zw + aw^2) = x^2 + dt_1(r^2 + rs + s^2b)$$

which has a solution in $E_1$ such that both sides are in $F_1$, and since $L/F_1$ is ramified, $x_0, y_0$ can be chosen such that the value of both sides is odd.

(3) Let $\psi_1 := [1, a + c] \perp t_1[1, a]$ and $\psi_2 = [1, b + c] \perp dt_1[1, b]$.

First of all, suppose that there exists a binary form $\beta$ over $F_1$ dominated up to a scalar by $\psi_1$ and $\psi_2$. We have to get a contradiction.
(a) Suppose that $\beta$ is diagonal. Modulo a scalar, we may suppose that
$\beta = \langle 1, s \rangle$. Over $E_1$, we have $(\psi_1)_{E_1} \cong \langle\langle a, t_1 \rangle\rangle]_{E_1}$ and $(\psi_2)_{E_1} \cong \langle\langle b, t_1 d \rangle\rangle]_{E_1}$.
Hence, $(\psi_1)_{E_1(\sqrt{s})} \sim 0 \sim (\psi_2)_{E_1(\sqrt{s})}$. This means that the quaternion algebras
$[a, t_1)$ and $[b, t_1 d)$ are split over $E_1(\sqrt{s})$. Consequently, there exist $k, l \in E_1$
such that $[a, t_1)_{E_1} \cong [k, s)_{E_1}$ and $[b, t_1 d)_{E_1} \cong [l, s)_{E_1}$, a contradiction since
the inseparable slot $s$ belongs to $F_1^*$.

(b) Suppose that $\beta$ is not diagonal. Modulo a scalar, we may suppose
$\beta = [1, s]$ for some $s \in F^*$. Reproducing the same arguments as in the case
(a), we get $k, l \in E_1^*$ such that $[a, t_1)_{E_1} \cong [s, k)_{E_1}$ and $[b, t_1 d)_{E_1} \cong [s, l)_{E_1}$, a
contradiction since the separable slot $s$ belongs to $F_1^*$.

The quaternion algebras $[a, t_1)$ and $[b, t_1 d)$ have a common slot over $E_1$, hence $[a, t_1) \otimes [b, t_1 d)$ is not a division algebra over $E_1$. Consequently, there exist $x, y, z \in F_1$ such that $[a, t_1) \otimes [b, t_1 d) \cong [c, x) \otimes [y, z)$. We have $\psi_1 \perp x\psi_2 \in I_q^2 F$ because this form has trivial Arf invariant. Moreover, its Clifford algebra has index $\le 2$. It follows from [10] that there exists a separable extension $E'/F_1$ such that $(\psi_1 \perp x\psi_2)_{E'} \sim 0$, i.e., $(\psi_1)_{E'} \cong (x\psi_2)_{E'}$.

(4) We have seen in (3) that there exist $u, v \in F_1^*$ such that over $L := F_1(\wp^{-1}(u))$ we have $(\psi_1)_L \cong v(\psi_2)_L$, but there does not exist a binary form over $F_1$ dominated up to a scalar by $\psi_1$ and $\psi_2$. Let $\psi = \psi_1 \perp t_2\psi_2 \in I_q^2 F_2$. Without loss of generality we may suppose $v = 1$. The form $\psi$ becomes hyperbolic over $F_2(\wp^{-1}(u), \sqrt{t_2})$, hence the index of the Clifford algebra of $\psi$ has index at most 4. Hence, there exists a biquaternion algebra $A$ over $F_2$ such that $[C(\psi)] = [A] \in \text{Br}_2(F_2)$. Suppose that $\psi$ dominates a form $\sigma$ of dimension 4 such that $\sigma \in \text{GP}_2(F_2)$ or $\sigma$ is similar to a quasi-Pfister form. Let $\overline{\sigma}_0$ and $\overline{\sigma}_1$ be the first and the second residue forms of $\sigma$ with respect to the $t_2$-adic valuation of $F_2$. Note that $\psi_1$ is the first residue form of $\psi$, and thus $\overline{\sigma}_0$ is dominated by $\psi_1$. Similarly, $\overline{\sigma}_1$ is dominated by $\psi_2$.

(a) Suppose that $\sigma \in \text{GP}_2(F_2)$. By Lemma 5.5, we have two cases:

(i) $(\overline{\sigma}_0 \in \text{GP}_2(F_1)$ and $\dim \overline{\sigma}_1 = 0$) or $(\overline{\sigma}_1 \in \text{GP}_2(F_1)$ and $\dim \overline{\sigma}_0 = 0)$. In the first case, $\overline{\sigma}_0 \cong \psi_1$, and thus $\psi_1$ has trivial Arf invariant, which is not possible because $c \notin \wp(F_1)$. Similarly, the case $\overline{\sigma}_1 \in \text{GP}_2(F_1)$ is not possible.

(ii) $\dim \overline{\sigma}_0 = \dim \overline{\sigma}_1 = 2$ and $\overline{\sigma}_0$ is similar to $\overline{\sigma}_1$. This is not possible since there does not exist a binary form which dominated up to a scalar by $\psi_1$ and $\psi_2$. This is not possible by (3).

(b) Suppose that $\sigma$ is similar to a quasi-Pfister from. By Lemma 5.6, $\overline{\sigma}_0$ and $\overline{\sigma}_1$ are similar of dimension 2. This implies that there exists a binary form which is dominated up to a scalar by both forms $\psi_1$ and $\psi_2$. This is not possible by (3).

(5) Let $A$ a biquaternion algebra over $F_2$ such that $[C(\psi)] = [A]$, where
$\psi = \psi_1 \perp t_2\psi_2$. Let $\alpha$ be an Albert form associated to $A$. Let $\varphi = \psi \perp t_3\alpha$.
We have $\dim \varphi = 14$ and $\varphi \in I_q^3 F_3$ since $\varphi$ has trivial Clifford invariant. We will prove that $\varphi$ does not contain a subform in $GP_2(F)$. In particular, $\varphi$ is not similar to the sum of two 3-fold Pfister forms, and thus the property $D(14)$ fails for $F_3$.

In fact, suppose that $\varphi$ contains a subform $\sigma \in GP_2(F)$. Let $\overline{\sigma_0}$ and $\overline{\sigma_1}$ be the first and the second residue forms of $\sigma$. We have $\overline{\sigma_0} \subset \psi$ and $\overline{\sigma_1} \subset \alpha$. We discuss on $\dim \overline{\sigma_1}$ using Lemma 5.5:

(i) Suppose that $\dim \overline{\sigma_1} = 4$. Then, $\overline{\sigma_1} \in GP_2(F_2)$, and thus $\alpha$ is isotropic, which is not possible.

(ii) Suppose that $\dim \overline{\sigma_1} = 2$. Then, $\overline{\sigma_0}$ is of dimension 2 and similar to $\overline{\sigma_1}$. Hence, there exists a quadratic extension $L/F_2$ for which $\psi_L$ and $\alpha_L$ are isotropic. It follows from Lemma 5.7 that $\psi$ dominates a form of dimension 4 which is in $GP_2(F_2)$ or similar to a quasi-Pfister form. This is not possible by (4).

(iii) Suppose that $\dim \overline{\sigma_1} = 0$. Then, $\psi$ contains as a subform $\overline{\sigma_0} \in GP_2(F_2)$, which is not possible by (4). $\square$

**Proof of Theorem 5.1.** By Theorem 5.4 there exists a field $F$ of characteristic 2 and scalars $a, b, c \in F$ such that $N_1(a, b, c) \neq 1$. We apply Proposition 5.10 to $F$ to get an extension $K$ of $F$ for which there exists $\varphi \in I_q^3 K$ anisotropic of dimension 14 such that $\varphi$ is not similar to the sum of two forms in $P_3 K$. Hence, $\ell_{2,14}(\varphi) > 2$. It follows from Theorem 4.8 that $\ell_{2,14}(\varphi) = 3$. $\square$

**Remark 5.11.** It is clear that $\ell_{1,d}(\varphi) = \frac{d}{2} - 1$ for any anisotropic quadratic form $\varphi \in I_q^2 F$ of dimension $d = 4$ or $6$. Now for $\varphi \in I_q^2 F$ of dimension 8 we have $\ell_{1,8}(\varphi) = 2$ if ind $C(\varphi) = 2$ and $\ell_{1,8}(\varphi) = 3$ if ind $C(\varphi) > 4$. However, when ind $C(\varphi) = 4$ we have $\ell_{1,8}(\varphi) \in \{2, 3\}$. The value 2 is obviously realized, and statement (4) of Proposition 5.10 provides an example of a field $k$ and a form in $I_q^2 k$ of dimension 8 which is a sum of two forms in $GP_2 k$.

## 6. BILINEAR FORMS

Our aim in this section is to provide some results on bilinear forms in $I^3 F$ of dimension less than or equal to 12 analogous to those of Pfister [25].

We recall that for bilinear forms we have the following isomorphism due to Kato $f^n : I^n F / I^{n+1} F \to \nu_F(n)$ given by:

$$f^n(\langle\langle a_1, \dots, a_n \rangle\rangle_b + I^{n+1}F) = \frac{da_1}{a_1} \wedge \dots \wedge \frac{da_n}{a_n},$$

where $\nu_F(n)$ is the additive group generated by pure $n$-logarithmic symbols $\frac{da_1}{a_1} \wedge \dots \wedge \frac{da_n}{a_n}$ with $a_1, \dots, a_n \in F^*$.
We define the symbol length of an element $\theta \in \nu_F(n)$ to be the smallest number of pure $n$-logarithmic symbols needed to express it. This number is denoted by $\ell'_n(\theta)$. As we did for quadratic forms, for any bilinear form $B \in I^n F$ of dimension $d$, we denote by $\ell'_{n,d}(B)$ the smallest number of forms in $GBP_n F$ needed to write $B$. Clearly, $\ell'_n(\theta) \le \ell'_{n,d}(B)$ when $f^n(B + I^{n+1}F) = \theta$. In this section we will restrict ourselves to the invariant $\ell'_{n,d}$ for $n=2$ or ($n=3$ and $d \le 12$). The case $n=2$ is an easy fact given in the following proposition. We include it to make the paper more self-contained.

**Proposition 6.1.** Let $B \in I^2 F$ be an anisotropic form of dimension $d$. Then, $\ell'_{2,d}(B) \le \frac{d}{2} - 1$.

*Proof.* Put $d = 2n + 2$ and $B = \langle 1, a_1, b_1, \dots, a_n, b_n, \prod_{i=1}^n a_i b_i \rangle_b$. We have $n \ge 1$ by the Hauptsatz. If $n=1$ then $B \in GBP_2 F$ and we are done. Suppose that $n > 1$ and write $B = C \perp D$, where $C = \langle 1, a_1, b_1, \dots, a_{n-1}, b_{n-1}, \prod_{i=1}^{n-1} a_i b_i \rangle_b \in I^2 F$ of dimension $d-2$, and $D = \langle a_n, b_n, \prod_{i=1}^{n-1} a_i b_i, \prod_{i=1}^n a_i b_i \rangle_b \in GBP_2 F$. By induction we have $\ell'_{2,d-2}(C) \le \frac{d-2}{2} - 1$. Hence, $\ell'_{2,d}(B) \le \frac{d-2}{2} = \frac{d}{2} - 1$. $\square$

The Arason-Pfister Hauptsatz is also true for bilinear forms.

**Proposition 6.2.** ([11, Th. 23.7], [20, Lemma 4.8]) Let $B \in I^n F$ be an anisotropic bilinear form. Hence, $\dim B \ge 2^n$. Moreover if, $\dim B = 2^n$, then $B \in GBP_n F$.

**Corollary 6.3.** For an anisotropic bilinear form $B \in I^3 F$ of dimension 8, we have $\ell'_{3,8}(B) = 1$.

Using Theorem 4.1, we obtain the following result for bilinear forms:

**Proposition 6.4.** Let $B \in I^n F$ be an anisotropic bilinear form. If $\dim B > 2^n$, then $\dim B \ge 2^n + 2^{n-1}$.

*Proof.* Let $\varphi = B \otimes [1, t^{-1}]$ be a nonsingular quadratic form over the rational function field $F(t)$. Since $B \in I^n F$, then $\varphi \in I_q^{n+1} F(t)$. We have $\dim \varphi > 2^{n+1}$. We deduce by Theorem 4.1 that $\dim \varphi \ge 2^{n+1} + 2^n$, which means that $\dim B \ge 2^n + 2^{n-1}$. $\square$

**Proposition 6.5.** Let $B \in I^3 F$ be of dimension 10. Then, $B$ is isotropic and there exists $\pi \in GBP_3 F$ such that $B \sim \pi$. In particular, $\ell'_{3,10}(B) \le 1$.

*Proof.* We combine Propositions 6.4 and 6.2. $\square$

For a non-zero totally singular quadratic form $Q = \langle a_1, \dots, a_n \rangle$, we associate the field $N_F(Q) := F^2(\alpha\beta \mid \alpha, \beta \in D_F(Q))$, called the norm field of $Q$. The degree $[N_F(Q) : F^2]$ is called the norm degree of $Q$ over $F$, denoted
by $ndeg_F(Q)$. We refer to [13, Section 8] for more details on norm fields of totally singular quadratic forms.

We recall a result on Witt kernel of a totally singular form for bilinear forms:

**Theorem 6.6.** [19, Theorem 1.2] Let $B$ be an anisotropic bilinear form and $Q$ an anisotropic totally singular quadratic form of norm degree $2^d$. Then, $B_{F(Q)}$ is metabolic if and only if $B \simeq \alpha_1\tau_1 \perp \cdots \perp \alpha_s\tau_s$, where $\alpha_1, \alpha_2, \dots, \alpha_s \in F^*$, $\tau_1, \dots, \tau_s \in BP_dF$ such that $\tilde{\tau}_i \simeq Q$ for all $1 \le i \le s$. In particular, $\dim B$ is divisible by $2^d$.

**Theorem 6.7.** [21, Proposition 3.8] Let $L = F^2(x_1, \dots, x_n)$ with $[L : F^2] = 2^n$, and $B, C$ are bilinear forms in $BP_nL$. Then, there exists $D \in BP_nL$ such that $B \perp C \sim D$.

For the classification of forms in $I^3F$ of dimension 12, we start by proving two lemmas.

**Lemma 6.8.** Let $B \in I^3F$ be an anisotropic bilinear form of dimension 12. Let $\tau \in BP_2F$ be an anisotropic bilinear form such that $B_{F(\tau)}$ is metabolic. Then, there exists $d \in F^*$ and $\theta$ an Albert bilinear form such that $B \simeq \langle 1, d \rangle_b \otimes \theta$. In particular, $\ell'_{3,12}(B) \le 2$

*Proof.* Let $Q$ be the totally singular quadratic form $\tilde{\tau}$. Since, $B_{F(Q)} \sim 0$ and $ndeg_F(Q) = 2^2$, we get by Theorem 6.6 $B \simeq \alpha_1\tau_1 \perp \alpha_2\tau_2 \perp \alpha_3\tau_3$, where $\alpha_i \in F^*$ and $\tau_i \in BP_2F$ with $\tilde{\tau}_i \simeq Q$ for all $1 \le i \le 3$. Since $B \in I^3F$, we get:

$$ (3) \qquad \tau_1 \perp \tau_2 \perp \tau_3 \in I^3F. $$

We extend this form to $F(\tau_3)$. Since $\dim (\tau_1 \perp \tau_2)_{\text{an}} \le 6$, we get by Theorem 6.6 that $i_W(\tau_1 \perp \tau_2) \ge 2$. By [21, Proposition 3.6], $\tau_1$ and $\tau_2$ are 1-linked. Hence, we can write $\tau_1 \simeq \langle\langle k, \ell \rangle\rangle$ and $\tau_2 \simeq \langle\langle k, m \rangle\rangle$ for some $k, \ell, m \in F^*$. We replace $\tau_1$ and $\tau_2$ in equation (3) to get:

$$ \ell \langle\langle k, \ell m \rangle\rangle \perp \tau_3 \in I^3F. $$

In particular, $\langle\langle k, \ell m \rangle\rangle \perp \tau_3 \in I^3F$, and by Hauptsatz we get:

$$ \tau_3 \simeq \langle\langle k, \ell m \rangle\rangle. $$

As a consequence, we have:

$$ B \simeq \langle\langle k \rangle\rangle_b \otimes (\langle\langle \alpha_1, \alpha_1 \ell, \alpha_2, \alpha_2 m, \alpha_3, \alpha_3 \ell m \rangle\rangle_b). $$

□

**Lemma 6.9.** Let $B \in I^3F$ be an anisotropic bilinear form of dimension 12 and $\tau \in BP_2F$ anisotropic such that $\dim (B_{F(\tau)})_{\text{an}} = 8$. Then, there exist $\lambda_i \in GBP_3F$ for $1 \le i \le 5$ such that $B \sim \perp_{1 \le i \le 5} \lambda_i$. In particular, $\ell'_{3,12}(B) \le 5$.
*Proof.* Suppose $x, y \in F^*$ such that $\tilde{\tau} \approx Q = \langle\langle x, y \rangle\rangle$. Since the extension $F(Q)/F$ is excellent for bilinear forms [22], there exists an $F$-bilinear form $D$ of dimension 8 such that $(B_{F(Q)})_{\text{an}} \approx D_{F(Q)}$. In particular, $D_{F(Q)} \in I^3F(Q)$. Moreover, $D + I^3F$ belongs to the Witt kernel $\ker(I^2F/I^3F \to I^2F(Q)/I^3F(Q))$. Using a result of Aravire-Baeza [3, Theorem 2.2], there exist $u, v \in F^2(x, y)$ such that $D \perp \langle\langle u, v \rangle\rangle_b \in I^3F$. Let $z \in D_F(D)$. Since $\dim(D \perp z \langle\langle u, v \rangle\rangle_b)_{\text{an}} < 12$, there exists $\pi \in GBP_3F$ (Proposition 6.5) such that $D \perp z \langle\langle u, v \rangle\rangle_b \sim \pi$. Passing to $F(Q)$, we get $D_{F(Q)} \sim \pi_{F(Q)}$ (as $(\langle\langle u, v \rangle\rangle_b)_{F(Q)}$ is metabolic as $u, v \in F^2(x, y)$ and $Q = \langle\langle x, y \rangle\rangle$). Since $(B_{F(Q)})_{\text{an}} \approx D_{F(Q)}$, we get $B_{F(Q)} \sim \pi_{F(Q)}$.

Now, $(B \perp \pi)_{F(Q)} \sim 0$ implies that $B \perp \pi \sim \perp_{1 \le i \le 5} \alpha_i \lambda_i$, where $\alpha_i \in F^*$ and $\lambda_i \in BP_2F$ such that $\tilde{\lambda}_i \approx Q$ (Theorem 6.6). Since $\perp_{1 \le i \le 5} \alpha_i \lambda_i \in I^3F$, hence $\perp_{1 \le i \le 5} \lambda_i \in I^3F$. Using Theorem 6.7, there exists $\theta \in BP_2F$ such that $\perp_{1 \le i \le 5} \lambda_i \sim \theta$. Since $\theta \in I^3F$, we get by Hauptsatz $\theta \sim 0$. Moreover, $\lambda_5 \sim \lambda_1 \perp \lambda_2 \perp \lambda_3 \perp \lambda_4$. Hence, $B \perp \pi \approx \perp_{1 \le i \le 4} (\alpha_i \lambda_i \perp \alpha_5 \lambda_i)$. Therefore, $\ell'_{3,12}(B) \le 5$. $\square$

**Theorem 6.10.** Let $B \in I^3F$ be anisotropic of dimension 12. Then, $\ell'_{3,12}(B) \le 5$.

*Proof.* We can write $B \approx \langle 1, x, y \rangle_b \perp \dots$ for some $x, y \in F^*$ up to a scalar. Let $Q \approx \langle 1, x, y, xy \rangle$. Since $\dim(B_{F(Q)})_{\text{an}} < 12$, using Proposition 6.5 we get $\dim(B_{F(Q)})_{\text{an}} \in \{0, 8\}$. If $B_{F(Q)} \sim 0$, we get the desired result by Lemma 6.8. If $\dim(B_{F(Q)})_{\text{an}} = 8$, we apply Lemma 6.9 to get the desired result. $\square$

## ACKNOWLEDGEMENT

This work was done during visits to the Academic College of Tel-Aviv-Yaffo in the Summer of 2023 and to CIRM in the summer of 2024 in the framework of the project *IEA of CNRS* between the Artois University and the Academic College of Tel-Aviv-Yaffo. The three authors are grateful for the support of both institutions and CNRS.

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SCHOOL OF COMPUTER SCIENCE, ACADEMIC COLLEGE OF TEL-AVIV-YAFFO, RABENU YERUHAM
St., P.O.B 8401 YAFFO, 6818211, ISRAEL

Email address: adam1chapman@yahoo.com

UNIV. ARTOIS, UR 2462, LABORATOIRE DE MATHÉMATIQUES DE LENS (LML), F-62300 LENS, FRANCE

Email address: ahmed.laghribi@univ-artois.fr

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ALBERTA, 11324 89 AVE NW, EDMONTON, AB T6G 2J5, CANADA

Email address: diksha.mukhija@gmail.com