# Formally real field

From Encyclopedia of Mathematics

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.

#### References

- 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

**How to Cite This Entry:**

Formally real field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Formally_real_field&oldid=54051