Alexander Sivatski: Nonexcellence of certain field extensions

AS3476.spb.edu

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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