Geodesic
Ordered fields and the ultrafilter theorem
Fundamenta Mathematicae, Tome 159 (1999) no. 3, pp. 231-241

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We prove that on the basis of ZF the ultrafilter theorem and the theorem of Artin-Schreier are equivalent. The latter says that every formally real field admits a total order. 
R. Berr; F. Delon; J. Schmid. Ordered fields and the ultrafilter theorem. Fundamenta Mathematicae, Tome 159 (1999) no. 3, pp. 231-241. doi: 10.4064/fm-159-3-231-241
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