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MULTIQUADRATIC EXTENSIONS, RIGID FIELDS AND PYTHAGOREAN FIELDS

Published online by Cambridge University Press:  15 March 2002

DAVID B. LEEP  and
TARA L. SMITH
DAVID B. LEEP
Affiliation:
Dept. of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027, USAleep@ms.uky.edu
TARA L. SMITH
Affiliation:
Dept. of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USAtsmith@math.uc.edu
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Abstract

Let F be a field of characteristic other than 2. Let F(2) denote the compositum over F of all quadratic extensions of F, let F(3) denote the compositum over F(2) of all quadratic extensions of F(2) that are Galois over F, and let F{3} denote the compositum over F(2) of all quadratic extensions of F(2). This paper shows that F(3) = F{3} if and only if F is a rigid field, and that F(3) = K(3) for some extension K of F if and only if F is Pythagorean and K = F(√−1). The proofs depend mainly on the behavior of quadratic forms over quadratic extensions, and the corresponding norm maps.

Type
PAPERS
Information
Bulletin of the London Mathematical Society , Volume 34 , Issue 2 , March 2002 , pp. 140 - 148
Copyright
© 2002 The London Mathematical Society

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