A field F is n-real if −1 is not the sum of n
squares in F. It is shown that a field F is m-real if and only
if rank (AAt
) = rank (A) for every n × m matrix A with entries
from F. An n-real field F is n-real closed if every proper algebraic
extension of F is not n-real. It is shown that if a 3-real field F
is 2-real closed, then F is a real closed field. For F a quadratic
extension of the field of rational numbers, the greatest integer n
such that F is n-real is determined
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