Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-08T15:19:30.104Z Has data issue: false hasContentIssue false
  • English
  • Français

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, [email protected]
TARA L. SMITH
Affiliation:
Dept. of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)