Recently I was led to consider some questions about Cremona transformations over arbitrary fields, which led deeper than planned into parts of algebraic geometry. I studied only algebraic groups in grad school, and read more by myself over the years with no intention of making this a focus, so my expertise is limited. Most of all I am trying to locate references to a few results which seem fundamental. But, to my surprise, diligent searches through older (Hodge and Pedoe, Semple and Roth, surveys edited by Snyder,...) and many later (mid 50s on) treatises, my favorites being more recent ones by Liu and Vakil, failed to turn up anything relevant, even in char 0.
Are the following obvious or well known -- and WHERE are they?:
(1) Any rational map $\phi: P^n_k \dashrightarrow P^n_k$ that becomes birational on extending to a larger field is already birational over k. In other words, a Cremona transformation and its inverse are always defined over the same subfields.
One way, maybe not the most direct, uses explicit inversion formulas for coeffs. It turns out that for any given degree (as below), independently of the field, there are a finite number of possible formulas. This simplifies and goes well beyond a result of Semple and Tyrrell (1968) on types over fields containing C. I am writing a short article about these ideas.
Somewhat related, but almost trivial (why is this not in standard texts?), is:
(2) The degree d of any $\phi: P^m_k \dashrightarrow P^n_k$ does not change under field extension.
By degree it is meant the least degree d of homogeneous polynomials that represent phi. This is not to be confused with a more common notion of degree, say for certain finite dominant morphisms, which often conflicts with the degree used here, e.g. when $\phi$ is birational. Equivalently, use the least d such that $\phi$ factors as the d-fold (Veronese) embedding of $P^m_k$ followed by a projection.
In terms of divisors, the problem is to show that over an algebraic closure K of k, the largest $D$ such that $\phi_K$ factors through $\mathcal L(-D)$ is definable over k. This has some connection with results of Chow (1950) obtained via Chow coordinates. There may be more relevant ideas in Weil's Foundations, which I found indigestible. Ignore this if you are not interested in history.
In terms of twisting sheaves, d seems to be the least number such that the image of $\mathcal O(1)$ under $\phi^*$ is embeddable in $\mathcal O(d)$. I have been told that using $\phi^*$ it is obvious that d does not change under field extension, but I did not get any extra insight into this by thinking in terms of Proj and sections of sheaves, even after consulting good sources such as Vakil's latest version.
(3) Is there a fairly simple proof along these lines, and what would be the key points?
The only way I know is to abandon sophisticated points of view and use a basic result whose origin/locations remain bafflingly elusive:
(4) If some multivariate polynomials over k have no common factor (of positive degree), the same is true after extending the field. [WHERE is this?]
This is of course easy (reduce to studying the effect on factorization of polynomials when a generator is added to the underlying field; there are 3 cases), and should appear in some older algebra book, any time after Steinitz (1910) on field theory. I am only mildly interested in refs to constructive proofs, which may be much older (over C) but would be more complicated and presumably use some sort of resultant.
Going beyond what motivated me, but obviously a natural question, is:
(5) Are there similar results with the domain replaced by something more general than an open set (the base-point-free locus of $\phi$) in the variety $P^m_k$?