Nonexcellence of multiquadratic field extensions

Let <f>k0</f> be a field, <f>chark0≠2</f>, <f>n&ges;2</f>, <f>a,b1,…,bn∈k0&ast;</f> such that the elements <f><ovl type="bar" STYLE="S">a</ovl>,<ovl type="bar" STYLE="S">b</ovl>1,…,<ovl type="bar" STYLE="S">b</ovl>n∈k0&ast;/k0&ast;2</f> are linearly independent. We show that there exists a field extension <f>F/k0</f> and an anisotropic 4-dimensional form <f>ϕ</f> over <f>F</f> such that <f>d±(ϕ)=a</f>, the form <f>ϕF(√ of b</rad>1,…,<rad>b</rad>n)</f> is isotropic and the form <f>(ϕF(<rad>b</rad>1,…,<rad>b1,…,√ of b</rad>n)</f> is isotropic and the form <f>(ϕF(<rad>b</rad>1,…,<rad>bn)</f> is isotropic and the form <f>(ϕF(√ of b</rad>1,…,<rad>b1,…,√ of bn))an</f> is not defined over <f>F</f>. [Copyright &y& Elsevier]