Corollary 1

Tag 06ZE, permanent. Cite it as [Bourbaki, Tag 06ZE].

Algebra, chapter VI, § 2. Ordered fields

— Every extension of an ordered field of odd finite degree admits the structure of an ordered extension.

Such an extension, being monogenous (V, p. 40, Th. 1), is isomorphic to $K[X]/(f)$ , where $f$ is an irreducible polynomial of odd degree. Then it is enough to show that $f$ changes sign in $K$ (Prop. 3), which follows immediately from Prop. 4.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

V, p. 40, Th. 1
Prop. 3
Prop. 4