Equivalence between the definitions of birationally trivial variety

Question

I was consulting Manin's book "Cubic forms algebra, geometry, arithmetic", and in the chapter about unirationality I stumbuled in the definition of "birationally trivial". This is given as following No proof of the equivalence is given so I was wondering where I can find one or if you could help me proving it.

Answer 1

One direction is just as easy to see in the general case when $X$ and $Y$ are birationally equivalent varieties.

Let $\phi : X \to Y$ be a birational map of varieties. The key point is to note that in your case $k(\mathbb{P}^n) = k(x_1,...,x_n)$ (i.e., a purely transcendental extension of $k$). Consider the morphism induced on fields, namely $$\phi^* : k(Y) \to k(X)$$ given by $\phi^*(f) = f \circ \phi$. If $\phi$ has a birational inverse $\psi$ it is easy to check that the above homomorphism of fields has $\psi^*$ as an inverse - hence is an isomorphism.

Now for the other direction is it easier if $X = \mathbb{P}^n$, if $k(Y)$ and $k(x_1,..,x_n)$ are isomorphic, then for each $x_i$ there must be a corresponding $y_i \in k(Y)$, and one can check the map from $Y \to \mathbb{A}^n$ $$a \to (y_1(a),..., y_n(a))$$ is birational (hint: find its inverse) - note this is only on an affine subset, but that's enough (hint: show $\mathbb{A}^n$ and $\mathbb{P}^n$ are birational).