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Constructing quaternionic fields

Published online by Cambridge University Press:  18 May 2009

Theresa P. Vaughan
Theresa P. Vaughan
Affiliation:
University of North Carolina at Greensboro, Greensboro, Nc 27412, USA
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Let K be a field of characteristic different from 2, and a, b quadratically independent elements of K. Put J= K(√a, √b). In [4], Jensen and Yui discuss the question of quaternionic (Q8) extensions of J, and give a survey of known results. In [8], Ware discusses (among other things) some general conditions for, and relations between, the existence of Q8 and D4 (dihedral) extensions of K. A general theorem of Witt [9] says that J will have a quaternionic extension J(√u) if and only if there exists a 3 × 3 matrix P over K such that PPt = diag(a, b, 1/ab), and an appropriate value for u is given in terms of the entries of P. The problem of actually finding P in a particular case is not trivial.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 1 , January 1992 , pp. 43 - 54
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

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