In Bourbaki's "Commutative Algebra", p. 386, the valuation of a ring (and later a field) is not required to be surjective. The same is true for the definition that Hartshorne uses in his "Algebraic Geometry". p. 39. On the contrary, Atiyah-MacDonald define a discrete valuation in p. 94 to be a surjective mapping onto $\mathbb{Z}$.
How does this discrepancy affect the existence of a uniformizer (local parameter in algebraic geometry context) for discrete valuation rings?
In Bourbaki e.g. p. 392, the reference to the uniformizer is almost axiomatic; i can not see any proof of existence. On the other hand, the existence in Atiyah-MacDonald follows immediately from surjectivity.