Karim Johannes Becher, David B. Leep: The Kaplansky Radical of a quadratic field extension

becher@maths.ucd.ie, leep@email.uky.edu

Submission: 2013, Aug 1

The radical of a field consists of all nonzero elements that are represented by every binary quadratic form representing 1. Here, the radical is studied in relation to local-global principles, and further in its behaviour under quadratic field extensions. In particular, an example of a quadratic field extension is constructed where the natural analogue to the square-class exact sequence for the radical fails to be exact. This disproves a conjecture of Kijima and Nishi.

2010 Mathematics Subject Classification: 11E04, 11E81, 12D15, 12F05, 13J05, 14H05

Keywords and Phrases: quadratic form, local-global principle, quasi-pythagorean field, func- tion field, power series field, quadratic field extension

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