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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2003, Issue 3, Pages 1–6 (Mi adm381)

RESEARCH ARTICLE

$N$ – real fields

Shalom Feigelstock

Department of Mathematics, Bar–Ilan University, Ramat Gan, Israel

Abstract: 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 $rank (AA^t)=rank (A)$ 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.

Keywords: $n$-real, $n$-real closed.

Full text: PDF file (180 kB)

Bibliographic databases:
MSC: 12D15
Revised: 23.10.2003
Language:

Citation: Shalom Feigelstock, “$N$ – real fields”, Algebra Discrete Math., 2003, no. 3, 1–6

Citation in format AMSBIB
\Bibitem{Fei03} \by Shalom~Feigelstock \paper $N$~-- real fields \jour Algebra Discrete Math. \yr 2003 \issue 3 \pages 1--6 \mathnet{http://mi.mathnet.ru/adm381} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2048637} \zmath{https://zbmath.org/?q=an:1122.12001}