$N$ – real fields

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 $\text{rank }(AA^t)=\text{rank }(A)$ for every $n\times m$ 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.