Submission: 2007, Apr 15
Consider towers of fields $F_1\subset F_2\subset F_3$, where $F_3/F_2$ is a quadratic extension and $F_2/F_1$ is an extension, which is either quadratic, or of odd degree, or purely transcendental of degree $1$. We construct numerous examples of the above types such that the extension $F_3/F_1$ is not $4$-excellent. Also we show that if $k$ is a field, $char\ k\not=2$ and $l/k$ is an arbitrary field extension of degree $4$, then there exists a field extension $F/k$ such that the extension $lF/F$ is not $4$-excellent.
2000 Mathematics Subject Classification: 11E04
Keywords and Phrases: Quadratic form, Pfister form, field extension, residue map
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