Formally real field

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2020 Mathematics Subject Classification: Primary: 12J15 [MSN][ZBL]

A field $F$ which is capable of being made an ordered field. The existence of such an order is equivalent to the property that $-1$ is not a sum of squares in $F$: this is the Artin--Schreier theorem. A real closed field is a formally real field for which no algebraic extension is formally real.


  • Tsit Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929
  • A. R. Rajwade, Squares, London Mathematical Society Lecture Note Series 171 Cambridge University Press (1993) ISBN 0-521-42668-5 Zbl 0785.11022
  • J.W. Milnor, D. Husemöller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer-Verlag (1973) ISBN 0-387-06009-X Zbl 0292.10016
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