3

I have been reading, and I am unsure what is meant by the following,

Choose $m \geq n$ positive integers. Let $F$ be an ambient field of rational functions in $n$ independent variables over $\mathbb{Q}(x_{n+1} , . . . , x_m)$.

Could someone please explain to me what is meant by this?

To be more specific, could you please describe what is meant by an ambient field of rational functions in $n$ independent variables. And also what is meant by $\mathbb{Q}(x_{n+1} , . . . , x_m)$. Just a brief description of what the elements are like would suffice thank you!

Thanks!

user93826
  • 389
  • 2
  • 10

1 Answers1

6

To start with the last question: $\mathbb{Q}(x_{n+1}, \dots, x_m)$ is the field of rational functions in the variables $x_{n+1}, \dots, x_m$. Its elements are of the form $f(x_{n+1}, \dots, x_m) / g(x_{n+1}, \dots, x_m)$ with $f, g \in \mathbb{Q}[x_{n+1}, \dots, x_m]$.

The "ambient field $F$ of a field $G$" is often used as an (informal) way of talking about a larger (and somehow easier) field $F$ in which everything takes place.

My guess is that in this case $F$ is $\mathbb{Q}(x_{n+1}, \dots, x_m)(x_1, \dots, x_n)$. That is, $F$ is just $\mathbb{Q}(x_1, \dots, x_{m})$, but it should be viewed as an extension of $\mathbb{Q}(x_{m+1}, \dots, x_n)$.

Magdiragdag
  • 15,049