Example (5)
Tag 04TE, permanent. Cite it as [Bourbaki, Tag 04TE].
Commutative Algebra, chapter VI, § 3. Valuations
The mapping P(X, Y) ↦ P(T, e^T) of C[X, Y] to C((T)) is injective (Functions of a real variable, Chapter IV, §2, Proposition 9) and therefore can be extended to an isomorphism of C(X, Y) onto a subfield of C((T)). The restriction to this subfield of the valuation on C((T)) defined in Example 3 defines a valuation on C(X, Y) which is improper on C, whose order group is Z and whose residue field is C.
Proposition 4 of no. 2 allows us to construct a valuation whose order group and residue field are given:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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