\titlea{\S25}{Leading Form and Degree Function}
\leftrunningtitle{Chapter III. Some Applications}
%\leftrunningtitle{Specialization of Quadratic and Symmetric
%Bilinear Forms}
\rightrunningtitle{\S25  Leading Form and Degree Function}

Let $k$ be a field of arbitrary characteristic and let $\varphi$ be a {\it form} over $k$, which is just as before understood to be a {\it nondegenerate quadratic} form. 

In case the characteristic of $k$ is different from $2$,  we alternatively interpret  $\varphi$ as a bilinear form. In fact we identify $\varphi$ with the bilinear form $\beta$  on $k^n$ ($n = \dim
\varphi$) which satisfies  $\varphi (x) = \beta (x, x)$. $\{$Thus $\beta = \frac{1}{2} B_{\varphi}$, where  
$B_{\varphi}$ is the bilinear form associated to $\varphi$.$\}$ In particular we make the identification
$$\langle a_1, \dots, a_n \rangle  =  [a_1, \dots, a_n] \qquad\qquad (a_i \in k^*).$$
One should bear in mind that with respect to this identification 
quadratic Pfister forms over $k$ correspond to bilinear Pfister forms over $k$.
\medskip

\noi{\it In the rest of this section we will always assume that $\varphi$ is strictly regular.}

\begtheorem{1.} Assume that $\varphi$ is not hyperbolic. Then the minimum of the dimensions of the kernel forms $\ker (\varphi \otimes L)$, where $L$ runs through all field extensions of $k$ with  $\varphi \otimes L \not\sim 0$, is a $2$-power $2^d$.
\endtheorem

\begproof{} If $\dim\varphi$ is odd (and thus $\Char K\ne 2$), this minimum is clearly $1$, and the statement holds with $d=0$.

Thus assume that $\dim\varphi$ is even. Let  $(K_r \, \vert \, 0 \leq r \leq h)$ be a generic splitting tower of $\varphi$. Since $\varphi$ is not hyperbolic we have $h\geq 1$.

The kernel form of   $\varphi \otimes K_{h-1}$ has height $1$. By \S20, Theorem~5 it is thus of the form 
$a \tau$ with $a \in K_{h-1}^*$ and $\tau$ a $d$-fold Pfister form over  $K_{h-1}$ for some
$d \geq 1$. By our generic splitting theory (cf. the end of \S9 or \S10, Theorem~4) it is clear that $2^d$ is the required minimum.\qed
\endproof

\begdefinition{1.} We call the number $d$, appearing in Theorem~1, the {\it degree}\Idx{degree}\Idx{111degfi@$\deg\varphi$} of the form $\varphi$ and denote it by $\deg \varphi$.  If $\varphi$ is hyperbolic, we set $\deg
\varphi = \infty$.
\enddefinition

According to the proof of Theorem~1, we have  $\deg \varphi = 0$ in case  $\dim
\varphi$ is odd  (and so  $\Char k \ne 2$), and  $\deg  \varphi \geq 1$ otherwise. An anisotropic $n$-fold Pfister form clearly has degree $n$.

The notion of degree of a form is closely connected to the notion of a ``leading form'', which we introduce next.

\begdefinition{2.} Let $\varphi$ be nondegenerate and let $(K_r \,
\vert \, 0 \leq r \leq h)$ be a generic splitting tower of $\varphi$.
\smallskip
\setitemindent{(a)}
\item{(a)}  We call any field extension of $k$, which is specialization equivalent to  $K_{h-1}$ over $k$, a {\it leading field for} $\varphi$.\Idx{leading field}
\smallskip

\item{(b)} Let $F$ be a leading field for $\varphi$. By \S23 the kernel form $\psi$ of $\varphi \otimes F$ is of the form $a \tau$ for $\dim\varphi$ even, and of the form $a \tau'$ with $a \in F^*$ and $\tau$ a Pfister form for $\dim\varphi$ odd. $\{$Recall: $\tau'$ denotes the pure part of $\tau$.$\}$  We call this Pfister form $\tau$ over $F$ the {\it leading form of $\varphi$ over $F$}.\Idx{leading form}
\enddefinition

From the proof of Theorem~1 it is clear that in even dimension the degree of $\varphi$ corresponds to the degree of any leading form $\tau$ of $\varphi$. In odd dimension there is a connection between the degree of $\tau$ and the degree of the even-dimensional form $\varphi \perp - d (\varphi)$, where 
$d(\varphi)$ denotes the signed determinant of $\varphi$, interpreted as a one-dimensional quadratic form, cf. \S2.


\begtheorem{2.} Assume that $\varphi$ is not split and that  $\dim\varphi$ is odd $($and so $\Char k\not=2$ since $\varphi$ is strictly regular$)$. Let $\tau$ be a leading form of $\varphi$. Then the forms $\tau$ and  $\varphi \, \perp  - d (\varphi)$ have the same degree.
\endtheorem

\begproof{} The form $\psi \colon = \varphi \, \perp  -d(\varphi)$ is not split. It has signed determinant\fonote{We write  $d(\psi)=1$ instead of $d(\psi) = \langle 1\rangle$.}
$d(\psi) = 1$. Let $F$ be a leading field for $\varphi$ and $E$ a leading field for $\psi$. We have
$$\varphi \otimes F \sim a \tau', \ \psi \otimes E \sim b \sigma,$$
where $a$ and $b$ are scalars, $\tau$ is the leading form of $\varphi$ and $\sigma$ is the leading form of $\psi$. Let $n \colon = \deg \tau$, $m \colon = \deg
\sigma = \deg \psi$. Since $d (\psi) = 1$ we have $d(\sigma) = 1$, and so
$m \geq 2$. Now $d(\tau') = \langle - 1 \rangle$, so that $d
(\varphi) \otimes F = d(\varphi \otimes F) = \langle - a \rangle$. Hence
$$\psi \otimes F  \sim a \tau' \perp \langle a \rangle  = a \tau.$$
Therefore the kernel form of $\psi \otimes F$ has dimension $2^n$ and it follows that $m \leq n$. On the other hand we have
$$\varphi \otimes E  \sim b \sigma  \perp  (d(\varphi) \otimes E).$$
The form on the right hand side is of dimension $2^m+1$. It cannot be split since $\sigma$ is anisotropic and has dimension $\geq 4$. From the generic splitting theory (see for instance \S4, Scholium~1) it follows that $2^m +1 \geq \dim (a \tau') = 2^n-1$, and so
$2^m +2 \geq 2^n$. Since  $m \leq n$ and $m \geq 2$, we must have $m = n$.\qed
\endproof

We now describe a situation where the leading form of $\varphi$ can be determined. 

\begtheorem{3}  {\rm (cf. $[$K$_5$, Th.6.3] for $\Char k \ne 2$).}  Let $n
\in {\Bbb N}$. Let $\varphi$ be an orthogonal sum $a \tau  \perp  \psi$ with $a \in k^*$, $\tau$ a Pfister form over $k$ of degree $n \geq 1$, and $\psi$ a form over $k$ with $\deg \psi
\geq n+1$. Let $F$ be a leading field for $\varphi$.
\smallskip
\setitemindent{(ii)}
\item{$(i)$} The leading form of $\varphi$ over $F$ is $\tau \otimes F$.
\smallskip

\item{$(ii)$} If $\deg \psi \geq n+2$, then $\varphi \otimes F$ has kernel form
$(a \tau) \otimes F$.
\endtheorem

\begproof{{\rm (as in  [K$_5$, p.88f])}.} $(i)$ We may assume that $\psi$ is not split. We will show that $\varphi$ has degree $n$.

Let $(L_i \, \vert \, 0 \leq i \leq e)$ be a generic splitting tower of $\psi$. Assume that $\tau \otimes L_e
\sim 0$. Let $s$ be maximal in\fonote{For $n \in {\Bbb N}$ we denote the set of numbers $0,
1, 2, \dots, n$ by $[0,n]$.} $[0, e-1]$ with $\tau \otimes L_s \not\sim 0$, so that $\tau
\otimes L_s$ is anisotropic. Let  $\psi_s$ be the kernel form of  $\psi \otimes
L_s$. Then  $\tau \otimes L_s (\psi_s) \sim 0$.  It then follows from the Norm Theorem and the Subform Theorem in the usual manner that 
$b \psi_s < \tau \otimes L_s$ for some $b \in L_s^*$. Thus, $\deg \psi = \deg \psi_s \leq n$, which contradicts our assumption about $\psi$. Therefore  $\tau \otimes L_e$ is anisotropic. Hence, 
$\varphi \otimes L_e$ has kernel form $(a \tau) \otimes L_e$,
and it follows that $\deg \varphi \leq n$.

Assume that $\varphi$ has degree  $m < n$. Let $(K_j \, \vert \;
0 \leq j \leq h)$ be a generic splitting tower of $\varphi$. Now  $\varphi \otimes K_{h-1}$ is of the form
$b \rho$ with $b \in K_{h-1}^*$ and $\rho$ a Pfister form of degree $m$ over $K_{h-1}$. We have
$$\psi \otimes K_{h-1}  \sim b \rho  \perp (-a) (\tau \otimes K_{h-1}).$$
The form on the right has dimension  $2^m +2^n < 2^{n+1}$. From  $\deg
\psi \geq n+1$ it follows that $\psi \otimes K_{h-1} \sim 0$.

We obtain  $b \rho \sim a (\tau \otimes K_{h-1})$. Since  $\dim \rho
< \dim \tau$, it follows that  $\tau \otimes K_{h-1} \sim 0$, and so
$\rho \sim 0$, a contradiction. Thus $\varphi$ has degree $n$. 

$(ii)$ We have $\psi \otimes K_h \sim (- a \tau) \otimes K_h$ and
$\deg \psi > n$. Thus $\psi \otimes K_h \sim 0$ and so $\tau
\otimes K_h \sim 0$.  Let $s$ be maximal in $[0, h-1]$ with $\tau \otimes
K_s \not\sim 0$, so that $\tau \otimes K_s$ is anisotropic.  From the Norm Theorem and the Subform Theorem it follows again that $b \varphi_s < \tau
\otimes K_s$ for some $b \in K_s^*$.  Since $\varphi$ has degree $n$ we must have $s = h-1$ and
$$\varphi_{h-1}  \cong  b (\tau \otimes K_{h-1}).$$
Thus $\tau \otimes K_{h-1}$ is the leading form of $\varphi$
over $K_{h-1}$. Furthermore,
$$\psi \otimes K_{h-1} \sim [ \varphi  \perp  (- a \tau)] \otimes K_{h-1}
\sim \langle b, -a \rangle \otimes \tau \otimes K_{h-1}.$$
If $\deg \psi \geq n+2$, then $\psi \otimes K_{h-1} \sim
0$ and so
$$\varphi_{h-1} \;\cong\; (a \tau) \otimes K_{h-1}.$$
\qed
\endproof

It is already clear from Definition~1 that every form which is Witt equivalent to $\varphi$ is of the same degree as $\varphi$. Therefore we have a degree function
$$\deg \colon \; \Wq (k) \longrightarrow {\Bbb N}_0 \cup \{ \infty \}$$
on the Witt group $\Wq (k)$ of strictly regular forms over $k$, 
given by $\deg (\{\varphi\}) \colon = \deg
\varphi$. In $\Char k = 2$ this function has values in ${\Bbb N}
\cup \{ \infty \}$.

The degree function motivates a further definition.

\begdefinition{3.} For every  $n \in {\Bbb N}$ we denote by $J_n (k)$\Idx{111jnk@ $J_n (k)$}
the set of all Witt classes $\xi$ of strictly regular forms with
$\deg (\xi) \geq n$.
\enddefinition

What does this notation mean for  $n = 1$? Clearly $J_1 (k)$ is the set 
of Witt classes of strictly regular forms of even dimension. If
$\Char k = 2$, we thus have $J_1 (k) = \Wq (k)$. If $\Char k \ne 2$,
then  $\Wq (k) = W(k)$ is a ring and $J_1 (k)$ is an ideal of
$W(k)$, namely the kernel of the dimension index 
$$\nu \colon W(k) \longrightarrow {\Bbb Z}/2, \quad \{\varphi\} \longmapsto \dim
\varphi + 2 {\Bbb Z},$$
which already made an appearance in  \S 2. 
This homomorphism  $\nu$ is also available in
characteristic 2. Its kernel is denoted by  $I(k)$ and is called 
the {\it fundamental ideal}\Idx{fundamental ideal}\Idx{111irr1@$I(k)$} 
of $W(k)$. Thus $W(k)/I(k) = {\Bbb Z}/2$. 
In $\Char k \ne 2$ we have $I(k) =
J_1 (k)$.

In more generality we have:

\begtheorem{4.} For every $n \in {\Bbb N}$, $J_n (k)$ is a $W(k)$-submodule of $\Wq (k)$. 
\endtheorem

\begproof{} (As in [K$_5$, p.89f].)

(a) We want to show that $J_n (k)$ is closed under addition in $\Wq
(k)$. We will prove the following equivalent statement:
for any two strictly regular forms $\varphi_1, \varphi_2$ over $k$ we have
$$\deg (\varphi_1  \perp  \varphi_2) \geq  \Min (\deg \varphi_1, \deg
\varphi_2). \leqno(\ast)$$
This statement is trivial when $\varphi_1$ or $\varphi_2$
has odd dimension, and also when   $\varphi_1  \perp
\varphi_2 \sim 0$. In the following we exclude these cases.

Let $n \colon = \deg (\varphi_1  \perp  \varphi_2)$. There exists a field extension
$L$ of $k$ such that
$$\ker (\varphi_1 \otimes L  \perp  \varphi_2 \otimes
L) = a \rho,$$
for some  $a \in L^*$ and a  Pfister form $\rho$ of degree $n$.
By definition of the degree function we have
$$\deg (\varphi_i \otimes L) \geq \deg \varphi_i \qquad (i = 1,2).$$
Therefore it suffices to prove the statement for the forms
$\widetilde{\varphi}_i \colon = \varphi_i \otimes L$ instead of
$\varphi_i$.
If $\deg \widetilde{\varphi}_2 > n$, then the Witt equivalence
$$\widetilde{\varphi}_1 \sim a \rho  \perp  (- \widetilde{\varphi}_2)$$
and  Theorem 3 give us that  $\widetilde{\varphi_1}$ has degree $n$.
Thus we always have
$$\Min (\deg \widetilde{\varphi}_1, \deg \widetilde{\varphi}_2) \leq n.$$

(b) $W(k)$ is additively generated by the one-dimensional bilinear forms 
$\langle a \rangle$. Since for every strictly regular form 
$\varphi$, the form $a \varphi$ is of the same degree as $\varphi$,
it is clear that every  $J_n (k)$ is stable under multiplication with elements from 
$W(k)$.\qed
\endproof 

An immediate consequence of Theorem~4 is

\begcorollary{1.}  Let $\varphi$ and $\psi$ be strictly regular forms over
$k$ with $\deg \varphi \ne \deg \psi$. Then
$$\deg (\varphi  \perp  \psi) = \Min (\deg \varphi, \deg \psi).$$
\endcorollary

It is commonplace to denote the $n$-th power of the ideal
$I(k)$ by  $I^n (k)$\Idx{111irr2@$I^n (k)$}  
(instead of  $I(k)^n$). It is also customary to leave out the curly brackets 
$\{ , \}$, which indicate that not the form itself, but rather its Witt class, is considered. For example, one writes $\gamma \in I^n (k)$ to indicate that the Witt class of a nondegenerate bilinear form  $\gamma$ is in $I^n (k)$. We will follow this custom.

The ideal $I(k)$ is additively generated by the bilinear forms $a
\langle 1, b \rangle$ where $a, b \in k^*$. Thus $I^n
(k)$ is additively generated by the forms $a \tau$ where $a \in k^*$ and $\tau$
is an $n$-fold bilinear  Pfister form which we may in addition assume to be anisotropic.
This is also true for $n = 0$, if we formally write  $I^0 (k) = W(k)$.

The $W(k)$-module $J_1 (k)$ is additively generated by the binary forms $a \left[ {1 \atop 1} \, {1 \atop b} \right]$. Thus the $W(k)$-module $I^n (k) J_1 (k)$ is additively generated by the forms $a
\tau$, where  $a \in k^*$ and $\tau$ is a  quadratic anisotropic Pfister form
of degree $n+1$. Thus, by  Theorem~4, every element of 
$I^n (k) J_1 (k)$ is of degree  $\geq n+1$.  Hence we obtain

\begcorollary{2.} For every  $n \in {\Bbb N}_0$ we have
$$I^n (k) J_1 (k) \subset J_{n+1} (k).$$
\endcorollary

Arason and Pfister already showed in  1970, without a generic splitting theory (which did not exist yet at the time), that in characteristic $\ne 2$ every anisotropic form 
$\varphi \ne 0$ which lies  in $I^n (k)$ is of dimension at least
$2^n$ (the so-called Arason-Pfister Hauptsatz  [AP]). 
The proof of Arason-Pfister can be modified in such a way that it also works in characteristic~2, but we will not elaborate that here. 
$\{$One should of course replace  $I^n (k)$ by $I^{n-1} (k) J_1
(k)$.$\}$ The statement of our Corollary~2 is equivalent to the Arason-Pfistert Hauptsatz, in every characteristic.

Next we wish to characterize the strictly regular even-dimensional forms 
 $\varphi$ of  degree~1  by employing the discriminant algebra
 $\Delta (\varphi)$ (cf. the beginning of \S 13). We begin with some comments on this invariant of 
$\varphi$.

If $L/k$ is a  quadratic separable field extension, there exists a generator 
 $\omega$ of $L/k$ with minimal equation  $\omega^2
+ \omega = a$, where  $1 + 4 a \in k^*$.  The norm form  $N_{L/k}
(x)$ of $L/k$ has the value matrix $\left[ {1 \atop
1} \, {1 \atop a} \right]$ with respect to  $1, \omega$.
If $\Char k \ne 2$ then  $1, 1+2
\omega$ is an orthogonal basis with respect to the norm form, which is thus of the form
$\langle 1, - 1 - 4 a \rangle$. Also, $(1+2
\omega)^2 = 1 + 4 a$.

All of this remains basically true for $a = 0$ when $L$ is equal to the split quadratic separable algebra
$k \times k$. In this case the norm form is hyperbolic instead of anisotropic as before.

This can all be verified easily and so we obtain straight away the following elementary lemma.

\beglemma{1.} The isomorphism classes of quadratic separable $k$-algebras $L$ are in one-one correspondence with the isometry classes of binary 
 $(=$ $2$-dimensional\,$)$ Pfister forms $\tau$ via $L = \Delta
(\tau)$, $\tau = N_{L/k}$.
\endlemma

We recall further that in characteristic $2$ the discriminant algebra $\Delta (\varphi)$ 
is just the Arf invariant  in disguise (see the beginning of \S13), while in characteristic $\ne 2$ it corresponds to the signed determinant $d (\varphi)$ since then
$\langle 1, - d(\varphi) \rangle$ is the binary Pfister form associated to
$\Delta (\varphi)$. Thus one verifies immediately:

\beglemma{2.} Let $\varphi_1$ and $\varphi_2$ be two regular forms of even dimension. 
\smallskip
\setitemindent{(a)}
\item{$(a)$} $\Delta (\varphi_1 \perp \varphi_2) = 1$ if and only if
$\Delta (\varphi_1) = \Delta (\varphi_2)$.
\smallskip

\item{$(b)$} $\Delta (\varphi_1 \perp \varphi_2) = \Delta
(\varphi_1)$ if and only if $\Delta (\varphi_2) = 1$.
\endlemma

\begremark{} These statements can be placed on a more conceptual footing by endowing the set of isomorphism classes of quadratic separable $k$-algebras with a multiplication, which turns it into an abelian group  $QS(k)$ of exponent~$2$, and observing that for regular forms $\varphi_1, \varphi_2$ of even dimension, $\Delta (\varphi_1 \perp
\varphi_2) = \Delta (\varphi_1) \cdot \Delta (\varphi_2)$, cf.
[B$_1$, Chap.IV], [B$_2$], [Kn$_3$, \S 10]. 
\endremark

\begtheorem{5.} Let $\varphi$ be a strictly regular form of even dimension. Let 
$\tau$ be the binary  Pfister form over $k$ with
$\Delta (\varphi) = \Delta (\tau)$ $($cf. Lemma~$1)$.
\smallskip
\setitemindent{(a)}
\item{$(a)$}  If $\Delta (\varphi) = 1$, then $\deg \varphi \geq 2$.
\smallskip

\item{$(b)$} If $\Delta (\varphi) \ne 1$ $($thus $\tau$ is anisotropic$)$, then
$\deg (\varphi) = 1$ and $\varphi \sim \tau \perp \psi$ for some
strictly regular form $\psi$ of degree $\geq 2$. Furthermore, if
$F$ is a leading field for $\varphi$, then $\tau \otimes F$ is the associated leading form of 
$\varphi$.
\endtheorem

\begproof{}  Let $F$ be a leading field for  $\varphi$, and $\rho$
the associated leading form of  $\varphi$. Thus $\ker (\varphi \otimes
F) = a \rho$ for some scalar  $a \in F^*$. If $\Delta (\varphi) =
1$, then  $\Delta (a \rho) = 1$. It follows that $\rho$, and thus also
 $\varphi$, is of degree $\geq 2$.

Assume now that $\Delta (\varphi) \ne 1$. By Lemma~2 the form
$\psi \colon = \varphi \perp \tau$ has discriminant  $\Delta (\psi)
= 1$. Thus $\deg \psi \geq 2$, by the first part of the proof. From
 $\varphi \sim - \tau \perp \psi$  and  Theorem~3 it follows that
$\varphi$ has degree~1 and leading form $\tau \otimes F$.\qed
\endproof

We note that only part $(a)$ of the Lemma was used. Part $(b)$ will be needed for the first time only in \S27.

It follows immediately from this Theorem that $J_2 (k)$ is the set of all
Witt classes  $\{ \varphi \}$ of regular forms  $\varphi$ with
$\dim \varphi$ even and $\Delta (\varphi) = 1$. By the remark after Lemma~2 it follows further that
the group $QS(k)$ is isomorphic to the additive group  $J_1 (k)/J_2(k)$.
We will not need this in the sequel, but it is nevertheless good to know.

We return to an examination of the modules $J_n (k)$ for arbitrary $n$.

\begtheorem{6.} Let  $\varphi$ be a strictly regular quadratic form 
and $\alpha$ a nondegenerate bilinear form. Assume that $\alpha$  has odd dimension. Then
$$\deg (\alpha \otimes \varphi) = \deg \varphi.$$
\endtheorem

\begproof{} This is clear when the dimension of  $\varphi$ is odd, and also 
when $\varphi \sim 0$. Therefore we will exclude those cases. Thus let
 $h(\varphi) > 0$ and $n \colon = \deg
\varphi \geq 1$. By  Theorem~4 we have $\deg (\alpha \otimes \varphi)
\geq n$.

We choose a leading field $F$ for  $\varphi$ and obtain
$\varphi \otimes F \sim  a \tau$ for some $a \in F^*$ and some
$n$-fold  anisotropic Pfister form $\tau$. We also consider some decomposition
$$\alpha  \cong  \langle b \rangle \perp b_1 \rho_1 \perp \dots \perp b_r \rho_r$$
where $b, b_1, \dots, b_r \in k^*$ and $\rho_1, \dots, \rho_r$ are binary bilinear Pfister forms.
Hence,
$$(\alpha \otimes \varphi) \otimes F  \sim  b \tau \perp b_1 (\rho_1
\otimes \tau) \perp \dots \perp  b_r (\rho_r \otimes \tau),$$
where we shortened $(\rho_i \otimes
F) \otimes \tau$ to $\rho_i \otimes \tau$.  The $\rho_i \otimes \tau$ are
$(n+1)$-fold  quadratic  Pfister forms. By  Theorem~4,  $b_1
(\rho_1 \otimes \tau) \perp \dots \perp b_r (\rho_r \otimes \tau)$
has degree  $\geq n+1$. By Theorem~3 (or Corollary~1),
$(\alpha \otimes \varphi) \otimes F$ has degree  $n$. It follows that $\deg
(\alpha \otimes \varphi) \leq n$ and so $\deg (\alpha \otimes
\varphi) = n$. \qed
\endproof

We record a much weaker, but still interesting, version of this theorem:

\begcorollary{} Let  $\alpha$ be a bilinear form of odd dimension and
$\varphi$ a quadratic strictly regular form with
 $\varphi\not\sim 0$. Then  $\alpha \otimes \varphi \not\sim 0$. 
\endcorollary 

\begtheorem{7.} For arbitrary $m \geq 1$, $n \geq 1$ we have 
$$I^m (k) J_n (k)  \subset  J_{m+n} (k).$$
\endtheorem

\begproof{}  (cf. [K$_5$, p.91]). Of course it suffices to show that
$I(k) J_n (k) \subset J_{n+1} (k)$.  Since $I(k)$ is generated by the forms
$\langle 1, a \rangle$ as an ideal, the theorem boils down to the following statement:
If $\varphi$ is a strictly regular quadratic form of even dimension over
$k$ which is not hyperbolic, then, for given $a \in k^*$, the degree of 
$\langle 1, a \rangle \otimes \varphi$ is larger than the degree of
$\varphi$.

We prove this statement by induction on $h(\varphi)$. The case 
$h(\varphi) = 1$ is clear. Let $h(\varphi) > 1$. If the form
$\alpha \colon = \langle 1, a \rangle \otimes
\varphi$ splits, the statement is true. Assume now that  $\alpha \not\sim
0$. We choose a leading field $F$ for $\alpha$. Let $\rho$
be the associated leading form of  $\alpha$ over $F$. Since $\deg
(\varphi \otimes F) \geq \deg \varphi$ it suffices to show that
$\deg \rho > \deg (\varphi \otimes F)$. We have
$h(\varphi \otimes F) \leq h (\varphi)$. If we replace  $k$ by $F$ and
$\varphi$ by $\varphi \otimes F$, we are reduced to the case
$$\langle 1, - a \rangle \otimes \varphi  \sim  \rho \leqno(1)$$
for some anisotropic Pfister form $\rho$. Now we must show that
$\deg \rho > \deg \varphi$.

Assume next that $\rho \otimes k(\varphi)$ is anisotropic. We have
$$\langle 1, - a \rangle \otimes (\varphi \otimes k (\varphi))  \sim 
\rho \otimes k(\varphi)$$
and
$\deg (\varphi \otimes k(\varphi)) = \deg \varphi$, $\deg
(\rho \otimes k(\varphi)) = \deg (\rho)$, $h(\varphi \otimes
k(\varphi)) = h(\varphi) -1$. Thus $\deg \rho > \deg \varphi$ by the induction hypothesis.

The remaining case is when 
$\rho \otimes k(\varphi)$ splits. By the Norm Theorem and the Subform Theorem we have
$$\rho \cong b \varphi \perp \zeta, \leqno(2)$$
for some $b \in k^*$ and some form $\zeta$ over $k$.
Assume that $\zeta = 0$. Then
$$\langle 1, - a, - b \rangle \otimes \varphi  \sim  0$$
by (1) and (2), contradicting the corollary to the previous theorem.
Hence $\zeta \ne 0$, and so $\dim \varphi < \dim \rho$. Since
$\rho$ is a Pfister form, it follows that $\deg \varphi < \deg \rho$.\qed
\endproof

Nowadays it is known that for every natural number $n$,
$$J_n (k)  =  I^n (k)$$
when $\Char k\ne 2$
[OVV], and
$$J_n (k)  =  I^{n-1} (k) \Wq (k)$$
when $\Char k=2$ [AB]. Thus, in Theorem~7 we have, {\it de facto}, $I^m (k) J_n (k) =
J_{n+m} (k)$.

The proofs of these statements are very dissimilar, and go far beyond the scope of this book. The case characteristic $\ne 2$ seems to be the more difficult one. 

