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\magnification 1200
\documentstyle{amsppt}
\expandafter\redefine\expandafter\enddemo\expandafter{\expandafter\hfill
\expandafter$\expandafter\square\expandafter$\enddemo}
\NoBlackBoxes
\topmatter
\title
THE WITT RING KERNEL FOR A FOURTH DEGREE FIELD EXTENSION
\endtitle

\author
A.S. Sivatski
\endauthor

\affil
Department of Mathematics \\
St.Petersburg Electrotechnical University \\
197376, St.Petersburg, Russia\\
\endaffil

\thanks
The work under this publication was partially supported by
Royal society Joint Project{\it "Quadratic forms and central simple algebras
under field extensions"} and
Emory University.
\endthanks

\keywords
Witt ring, Pfister form, polynomial
\endkeywords

\def\phi{\varphi}
\def\a{\alpha}
\def\ind{\operatorname{ind}}
\def\la{\lambda}
\def\dim{\operatorname{dim}}
\def\det{\operatorname{det}}
\def\t{\theta}
\def\mod{\operatorname{mod}}
\def\disc{\operatorname{disc}}
\def\char{\operatorname{char}}
\def\Br{\operatorname{Br}}
\def\cd{\operatorname{cd}}
\def\ker{\operatorname{ker}}
\def\l{\langle}
\def\sep{\operatorname{sep}}
\def\r{\rangle}
\def\ll{\langle\!\langle}
\def\rr{\rangle\!\rangle}
\def\s{\sum\limits_{i=1}^n}
\def\o{\omega}
\def\w{\widetilde}
\def\z{\Bbb Z/2\Bbb Z}
\def\ss{\sum\limits_{j=1}^k}
\def\m{\matrix}
\def\e{\endmatrix}
\def\si{\sigma}
\def\mx{\left (\m 1&0\cr x&1\cr\e\right )}
\def\mxx{\left (\m 1&0\cr -x&1\cr\e\right )}
\def\ar{\Longrightarrow}

\abstract
We compute the Witt ring kernel for an
arbitrary field extension of degree 4 and
characteristic different from 2 in terms
of the coefficients of a polynomial determining the extension.
In the case where the lower field is not formally real we prove that
the intersection of any power n of its fundamental ideal
and the Witt ring kernel is generated by n-fold
Pfister forms.

\endabstract

%\subjclass 13C12, 13F20, (13F10, 13D40)
%\endsubjclass

\endtopmatter

\document

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\document

Let $F$ be a field of characteristic different from $2$. As usual denote
by $W(F)$ the Witt ring of $F$, i.e. the ring of equivalence classes
of quadratic forms over $F$.
In this note we compute the kernel
$W(L/F)$ of the restriction map $W(F)\to W(L)$,
where $L/F$ is an arbitrary field extension of degree $4$.
This kernel was computed in [2] in the case where $L/F$ is a tower of two
quadratic extensions, and our result can be considered as its generalization.
The main step is to show that $W(L/F)$ is generated as an ideal by $1$-fold
and $2$-fold Pfister forms. The last statement
is a particular case
of a stronger result (see Corollarry $2$), which
as far as we know, has not been known even for
biquadratic extensions.

Our notation is standard, and the main reference is the book [4]. Notice only
that all the
fields considered below are of characteristic different from $2$.
We write $\phi\simeq\psi$ if quadratic forms $\phi$ and $\psi$
are isomorphic, and $\phi =\psi$ if their classes in the Witt ring are equal.
The anisotropic part of a form $\phi$ is denoted by $\phi_{an}$.
We use the sign $\perp$ for the direct sum of two forms and the signs
$+$ and $-$ for the sum and the difference of elements of the Witt ring.
For a form $\phi$ over
a field $F$ we denote by $D_F(\phi )$ the set of nonzero values
of $F$ represented by $\phi$. By the $n$-fold Pfister form
$\ll a_1,\dots ,a_n\rr$ we mean the product
$\l 1,-a_1\r\otimes\dots\otimes\l 1,-a_n\r$.
If $V$ is a linear space over a field $F$ and
$L/F$ is a field extension, then by definition $V_L=L\otimes_F V$.

We begin by the following statement, which plays the main role in the sequel.
\proclaim{Proposition 1}
Suppose $p(t)$ is an irreducible even degree polynomial over a field $F$, and
$\t$ is its root.
Let $k$ be a nonnegative integer,
$\phi$ an anisotropic quadratic form over $F$ such that
$\dim\phi \ge {k\over 2}\deg p +1$ and $\dim (\phi_{F(\t)})_{an}=k$.
Then there exists a subform $\phi_0\subset\phi$ and $x\in F^*$ such that
$\dim\phi_0\le {1\over 2}\deg p +1$ and $xp(t)\in D({\phi_0}_{F(t)})$.
In particular, the form ${\phi_0}_{F(\t)}$ is isotropic.
\endproclaim
\demo{Proof}
Let $U$ be the linear space associated to $\phi$,
$U=\l e_1,\dots ,e_{\dim\phi}\r$. Let further
$V=\l v_1,\dots ,v_{{1\over 2}(\dim\phi -k)}\r$ be a totally
isotropic subspace of
$U_{F(\t)}$. Put
$W=\sum\limits_{i=0}^{{1\over 2}\deg p}\t^i U$.
We can consider $V$ and $W$ as $F$- linear subspaces of the $F$- linear space
$U_{F(\t)}$. Obviously,
$$\dim_F U_{F(\t)}=\dim\phi\deg p,\
\dim_F V={1\over 2}(\dim\phi -k)\deg p,\
\dim_F W=({1\over 2}\deg p +1)\dim\phi .$$
Since $\dim\phi >{1\over 2}k\deg p$, we have
$$\dim_F V +\dim_F W >\dim_F U_{F(\t)}.$$
Hence $V\cap W\not=0$. Let
$0\not= v= \sum\limits_{i=0}^{{1\over 2}\deg p}\t^iu_i\in V$, where
$u_i\in U$. Denote by $\phi_0$  the subform of $\phi$ determined by
the $F$-linear subspace $U_0=\l u_0,\dots u_{{1\over 2}\deg p}\r\subset U$.
Obviously, $\dim\phi_0\le {1\over 2}\deg p+1$.
Put
$\w v= \sum\limits_{i=0}^{{1\over 2}\deg p}t^iu_i\in F[t]\otimes_F U_0$.
Since $\phi (v)=0$, we have $p\vert \phi_0(\w v)$.
Since $\deg\phi_0 (\w v)\le\deg p$, we get that
$\phi_0(\w v)=xp$ for some $x\in F$. Finally, $x\not=0$, because $v\not=0$,
and $\phi$ is anisotropic over $F$.
\enddemo
We apply Proposition $1$ to field extensions of degree $4$.
\proclaim{Corollary 2}
Under hypothesis of Proposition $1$ suppose $\deg p=4$.
Then $\phi =\psi +\sum\pi_i$, where
$\dim\psi\le 2k$, each $\pi_i$ is similar to either a $2$-fold or a $1$-fold
Pfister form and ${\pi_i}_{F(\t)}=0$. Moreover,
we can assume that each $\pi_i$
is similar to a $2$-fold Pfister form except the case
where $k=0$, the extension $F(\t)/F$
contains a quadratic subextension $F(\sqrt d)/F$
such that $\phi_{F(\sqrt d)}=0$, and $\disc (\phi)=d$.
 In this case we can assume that exactly one $\pi_i$ is similar to
a $1$-fold Pfister form $($ obviously, then $\disc (\pi_i)=d)$.

In particular, if $k=0$, i.e. $\phi\in W(F(\t)/F)$, we have
$\psi =0$, and $\phi =\sum\pi_i$.
\endproclaim
\demo{Proof}
Assume first that $k\ge 1$.
By Proposition $1$ we get
$\phi\simeq\phi_0\perp\phi_1$, where $\dim\phi_0=3$ and ${\phi_0}_{F(\t)}$
is isotropic. Let $\pi\simeq\phi_0\perp\det (\phi_0)$.
Then
$$\phi =\phi_1 +\l-\det (\phi_0)\r +\pi ,$$
$\pi$ is similar to a $2$-fold Pfister form, $\pi_{F(\t)}=0$, and
$$\dim(\phi_1+\l -\det (\phi_0)\r)_{an}\le\dim\phi -2.$$
Therefore, we can go on by
induction on $\dim\phi$. In the case $k=0$ and $\dim\phi\ge 3$ the same
argument works. Finally, if $k=0$ and $\dim\phi =2$, then $\phi$ is similar
to a $1$-fold Pfister form. It is obvious that in this case the field $F(\t)$
contains the field $F(\sqrt{\disc\phi})$.
\enddemo
\smallskip
{\bf Remark.} Let us call a form $\phi$ minimal with respect to
an extension $F(\t)/F$, if $\dim\psi\ge\dim\phi$
for any form $\psi$ such that
$\phi -\psi\in W(F(\t)/F)$.
Corollary $2$ claims that if $\deg p=4$,
$\dim\phi >2k$ and $\dim (\phi_{F(\t)})_{an}=k$,
then $\phi$ is not minimal. On the other hand,
it is easy to see that if forms
$\phi_1$ and $\phi_2$ over $F$
are minimal with respect to the extension
$F(\t)/F$, then the form $\phi_1\perp x\phi_2$
over $F((x))$ is minimal
with respect to the extension
$F((x))(\t)/F((x))$. Let $\tau$ be a $4$-dimensional form over $F$
such that
the form $\tau_{F(\t)}$ is isotropic, but for any subform $\tau'\subset\tau$
of codimension $1$ the form $\tau'_{F(\t)}$ is anisotropic. (The existence
of such an extension $F(\t)/F$
of degree $4$ and a form $\tau$ has been established in [1] and [5]).
In particular, $\tau$ is minimal. Therefore, the forms
$\psi_1\simeq\tau\otimes\l x_1,\dots ,x_m\r$ and
$\psi_2\simeq\tau\otimes\l x_1,\dots ,x_m\r\perp\l x_{m+1}\r$ over the field
$K=F((x_1))((x_2))\dots ((x_{m+1}))$ are minimal, which shows that
the inequality $\dim\psi\le 2k$ in Corollary $2$ can not be
improved. Therefore, in the case $\deg p=4$
the inequality $\dim\phi \ge {k\over 2}\deg p +1$ in
Proposition $1$ can not be improved as well.
\smallskip
It remains to determine $2$-fold Pfister forms in $W(F(\t)/F)$.
This can be done in terms of the coefficients of $p$.
\proclaim{Lemma 3}
Let $p(t)=t^4 +at^2+bt+c$ be an irreducible polynomial over $F$,
$\t$ its root, $\pi$ an
anisotropic $2$-fold Pfister form over $F$.
Suppose that $\pi_{F(\sqrt d)}\not=0$ for any $d\in F^*$ such that
$F(\sqrt d)\subset F(\t)$.
Then the following two conditions are
equivalent:

$1)$\ $\pi_{F(\t)}=0$.

$2)$\ there exist $x,y\in F^*$ such that
$\det \left (\m -x& 0 &{1\over 2}(y-ax)\cr
           0& -y & -{1\over 2}bx\cr
           {1\over 2}(y-ax) & -{1\over 2}bx & -cx\cr\e\right )\in {F^*}^2$
and $\pi\simeq \ll x, y\rr$.
\endproclaim
\demo{Proof}
$1)\ar 2)$. Assume that $\pi_{F(\t)}=0$. Let $(V,\pi')$
be the quadratic space of the
pure subform of $\pi$, i.e. $\pi\simeq\l 1\r\perp\pi'$.
All $3$-dimensional subforms of $\pi$
are similar to each other. Therefore,
by Proposition $1$ there are $x\in F^*$ and
$v_0,v_1,v_2\in V$ such that
$$\pi' (t^2v_2+tv_1+v_0)=-xp.$$
We have $\phi_{F(\t)}\not=0$ for any
$2$-dimensional subform $\phi\subset\pi'$. Indeed,
otherwise we would have $F(\sqrt{\disc\phi} )\subset F(\t)$,
hence $\pi_{F(\sqrt{\disc\phi })}=0$,
a contradiction to the
hypothesis of the proposition. Therefore, the vectors
$v_0,v_1,v_2$ are linearly independent.
Comparing the coefficients at the powers of $t$ on both sides of the last
equality we get
$$\cases\pi' (v_2,v_2)=-x,\cr
\pi' (v_1,v_2)=0,\cr
\pi' (v_1,v_1)+2\pi' (v_0,v_2)=-ax,\cr
2\pi' (v_0,v_1)=-bx,\cr
\pi' (v_0,v_0)=-cx.\endcases$$
Let $\pi' (v_0,v_2)={1\over 2}(y-ax)$. Then the matrix of the form $\pi'$
with respect to the basis $(v_2,v_1,v_0)$ is
$\left (\m -x& 0 &{1\over 2}(y-ax)\cr
           0& -y & -{1\over 2}bx\cr
           {1\over 2}(y-ax) & -{1\over 2}bx & -cx\cr\e\right )$.
Since $\det\pi'=1$, we get $\pi'\simeq\l -x,-y,xy\r$, hence
$\pi\simeq\ll x,y\rr$.

$2)\ar 1)$.
We have $\pi' (t^2v_2+tv_1+v_0)=-xp$ for some vectors $v_0,v_1,v_2\in V$.
In particular, $\pi'_{F(\t)}$ is isotropic, which means that $\pi_{F(\t)}=0$.
\enddemo

\proclaim{Corollary 4}
Suppose a polynomial $p(t)=t^4+at^2+bt+c$ is irreducible
over a field $F$, and
$\pi$ is an anisotropic $2$-fold Pfister form over $F$ such that
$\pi_{F(\t)}=0$. Then at least one of the following two conditions hold:

$1)$ $\pi\simeq\ll d,e\rr$, where $F(\sqrt d)\subset F(\t)$, $e\in F^*$.

$2)$  $\pi\simeq\ll \alpha{(\alpha -a)}^2-4c\alpha +b^2,-\alpha\rr$
where $\alpha\in F$, and
$(\alpha{(\alpha -a)}^2-4c\alpha +b^2)\alpha\not=0$.

Conversely, if $\pi$ is a $2$-fold Pfister form of one of two types above,
then $\pi_{F(\t)}=0$.
\endproclaim
\demo{Proof}
Assume that $\pi_{F(\t)}=0$,
and condition $1)$ does not hold. Then, as we have noticed already,
 the vectors $v_0,v_1,v_2$ introduced in
Lemma $3$ are linearly independent. Put $\alpha ={y\over x}$.
Since
$$\det \left (\m -x& 0 &{1\over 2}(y-ax)\cr
           0& -y & -{1\over 2}bx\cr
           {1\over 2}(y-ax) & -{1\over 2}bx & -cx\cr\e\right )
=x({1\over 4}\alpha {(\alpha -a)}^2-\alpha c+{1\over 4}b^2)\in {F^*}^2,$$
and $\ll x,-x\rr =0$,
we have
$$\pi\simeq\ll x,y\rr\simeq\ll x,\alpha x\rr\simeq\ll x,-\alpha\rr =
\ll \alpha {(\alpha -a)}^2-4c\alpha +b^2,-\alpha\rr .$$

Conversely, if $(\alpha{(\alpha -a)}^2-4c\alpha +b^2)\alpha\not=0$,
then put $x= \alpha {(\alpha -a)}^2-4c\alpha +b^2$, $y=x\alpha$.
The implication $2)\Longrightarrow 1)$ of Lemma $3$ shows that
$$\ll\alpha {(\alpha -a)}^2-4c\alpha +b^2, -\alpha\rr_{F(\t)}=0.$$
\enddemo
{\bf Remark.} The polynomial $f(\alpha)=\alpha {(\alpha -a)}^2-4c\alpha +b^2$
is a cubic resolvent of the polynomial $p$.
\smallskip
{\bf Example.}
Suppose $a\in F^*\setminus {F^*}^2$, $b,c\in F$,
$b+2c\sqrt a\notin {F(\sqrt a )}^2$.
Consider the quartic extension $F(\sqrt{b+2c\sqrt a})/F$. If
$\t=\sqrt{b+2c\sqrt a}$, then
$$p(\t)=\t^4-2b\t^2+(b^2-4ac^2)=0.$$
Let $\pi$ be a $2$-fold Pfister form such that $\pi_{F(\t)}=0$,
$\pi_{F(\sqrt a)}\not=0$.
By Corollary $4$ we get that
$$\pi\simeq\ll \alpha{(\alpha +2b)}^2 -4(b^2-4ac^2)\alpha, -\alpha\rr\simeq
\ll {(\alpha +2b)}^2 -4(b^2-4ac^2), -\alpha\rr\simeq
\ll ae^2+be+c^2, -e\rr ,$$
where $e={{4c^2}\over\alpha}$.
In fact we have got the result of theorem $3.9$ from [1].

\proclaim{Proposition 5}
Let $L/F$ be a field extension of degree $4$, and $F$ is not formally real.
Then for any $n$
$$W(L/F)\cap I^n(F)=(W(L/F)\cap I^2(F))I^{n-2}(F).$$
Hence in terminology of [1] the ideal $W(L/F)\cap I^n(F)$ is an $n$-Pfister
ideal.
\endproclaim
\demo{Proof}
For any field $K$ denote by $G_K$ its absolute Galois group.
Unfortunately the proof depends on a very deep result of Voevodsky, which
claims that there exists an isomorphism
$I^n/I^{n+1}(K)\simeq H^n(G_K,\z )$, which takes an $n$-fold Pfister form
$\ll a_1,\dots ,a_n\rr$
to the cup product
$(a_1)\cup\dots\cup (a_n)$ ([6], [7]).
First we need a couple of lemmas.
\proclaim{Lemma 6}
Let $F$ be a field. The following two conditions are equivalent:

$1)$ $F$ is not formally real.

$2)$ There exists a subfield $F_1\subset F$ and a positive
integer $N$ such that $I^N(F_1)=0$.
\endproclaim
\demo{Proof}
$2)\Longrightarrow 1)$. If $F$ is formally real, then so is any subfield $F_1$
of $F$, hence $I^N(F_1)\not=0$ for any $N$.

$1)\Longrightarrow 2)$. Assume that $F$ is not formally real, and $s(F)=m=2^k$
is its level, i.e. the minimal number $m$ such that there exist
$x_1,\dots ,x_m\in F$ with the equality $x_1^2+\dots +x_m^2=-1$.
 Let $F_0$ be the prime subfield
of $F$, i.e. $F_0$ is either $\Bbb Q$ or $\Bbb Z/p\Bbb Z$.
Put $F_1=F_0(x_1,\dots ,x_m)\subset F$, and $N=m+k+3$.
Let us show that $I^N(F_1)=0$.
Since $\cd_2 (F_0(\sqrt{-1}))\le 2$,
we have $\cd_2 (F_1(\sqrt{-1}))\le m+2$, hence
$I^{m+3}(F_1(\sqrt{-1}))=0$.
 Therefore, for any $a_1,\dots ,a_{m+3}\in F_1^*$ we get
$$\ll a_1,\dots a_{m+3}\rr\simeq\ll -1,b_1,\dots ,b_{m+2}\rr$$
for some
$b_1,\dots b_{m+2}\in F_1^*$. From this it follows at once that for any
$a_1,\dots a_N\in F_1^*$
$$\ll a_1,\dots a_N\rr\simeq\ll -1\rr ^{\otimes {(k+1)}}
\otimes\ll b_1,\dots ,b_{N-k-1}\rr$$
for some $b_1 ,\dots , b_{N-k-1}\in F_1^*$.
On the other hand, since
$x_1,\dots x_m\in F_1$, we have
$\ll -1\rr ^{\otimes {(k+1)}}=0\in W(F_1)$.
This implies that
$\ll a_1,\dots ,a_N\rr=0\in I^N(F_1)$, which finishes the proof.
\enddemo
\proclaim{Lemma 7}
The kernel of the restriction map $I^2/I^3(F)\to I^2/I^3(L)$ is generated
by $2$-fold Pfister forms belonging to $W(L/F)$.
\endproclaim
\demo{Proof}
Assume that
$\a\in\ker (I^2/I^3(F)\to I^2/I^3(L))=\ker (_2\Br (F)\to _2\Br (L))$.
Obviously, $\ind\a\le 4$. Let $\phi$ be an Albert form associated to $\a$.
Then $\phi_L\in I^3(L)$. Since $\dim\phi =6$, we conclude that
$\phi\in W(L/F)$. In view of Corollary $2$ the lemma is proved.
\enddemo

We return to the proof of Proposition $5$.
It is proven in $[3]$ (cor.$18$ and the remark after conjecture $20$)
that the kernel of the restriction map of the graded rings
$H^*(G_F,\z )\to H^*(G_L,\z )$ is generated in degrees $1$ and $2$
as an ideal in $H^*(G_F,\z)$.
Let $\a\in W(L/F)\cap I^n(F)$.
Assume that the conditions of Lemma $6$ are fulfilled. Then
it is easy to see that there is an intermediate subfield
$F_1\subset\w F\subset F$
such that $\a\in W(L\w F/\w F)\cap I^n(\w F)$ and $\cd_2 (\w F)< \infty$.
Interpreting the graded ring
$H^*(G_{\w F},\z)$ (resp. $H^*(G_{L\w F},\z)$) as the graded ring
$I^*/I^{*+1}(\w F)$ (resp. $I^*/I^{*+1}(L\w F)$)
and using Lemma $7$ we can prove Proposition
$5$ straightforwardly by decreasing induction on $n$.
\smallskip
{\bf Open question.}
Does Proposition $5$ remain valid for a formally real field $F$ ?
\smallskip
{\bf Acknowledgements.} This research has been carried out at
Emory University, Atlanta and the University of Nottingham
in the framework of Royal society Joint Project
{\it "Quadratic forms and central simple algebras
under field extensions"}.
The author expresses his thanks to Detlev Hoffmann,
Raman Parimala and Alexander Vishik for very fruitful discussions.
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\frenchspacing

\refstyle{D}
\Refs\nofrills{REFERENCES}
\widestnumber\key{10}

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\jour preprint available at http://www.math.uiuc.edu/K-theory/0589/
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\ref
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\by Scharlau W.
\paper Quadratic and Hermitian forms
\jour Springer, Berlin Heidelberg New York
\yr 1985
\endref

\ref
\key 5
\by Sivatski A.S.
\paper Nonexcellence of certain field extensions
\jour Journal of Mathematical Sciences
\yr 2007
\vol 145
\issue 1
\pages 4811-4817
\endref

\ref
\key 6
\by Voevodsky V.
\paper The Milnor Conjecture
\jour preprint
\yr 1996
\endref

\ref
\key 7
\by Voevodsky V.
\paper Motivic cohomology with $\Bbb Z/2$ coefficients
\jour Publ. Math. Inst. Hautes \'Etudes Sci.
\yr 2003
\vol 98
\pages 59-104
\endref
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