# Q is the prime subfield of any field of characteristic 0, proof that

## Primary tabs

Defines:
prime field
Type of Math Object:
Proof
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

### A little sketchy

I think this entry confuses a couple of important points, stemming from the fact that you've tried to make the notion of a "ground field" halfway between an intuitive definition and a formal definition.

The formal counterpart to what you're talking about is the prime field, and your two theorems refer to those, Q and F_p (which, incidentally, is a better notation since Z_p can be confused with the p-adic integers).

The intuituve counterpart turns out to be not so intuitive after all..consider the scenario of considering the field extension \mathbb{C}(\sqrt{x}) over \mathbb{C}(x). One wouldn't really want to call Q the "ground field" in such a scenario, but further, nor is this notion really well defined. For example, \mathbb{C}(x) is just a degree 2 extension of \mathbb{C}(x^2), which in turn is a degree 2 extension of \mathbb{C}(x^4), etc. In this situation, there is no unique smallest subfield of the right type (i.e. a function field over \mathbb{C}) that embeds into all of the field in question.

I recommend either giving a formal definition of a prime field, or re-write to be clear that "ground field" is not a formal term, and that the expression is used informally by mathematicians to refer to an obvious choice of an important field lying around somewhere.

Cam

### Re: A little sketchy

NB, in some contexts one speaks also of "base field" (see e.g. the entry "extension field").
Jussi

### Re: A little sketchy

Thank you for your comment! Sorry to have caused this confusion. I have changed the title and the content of the entry. I have also added a request for someone to define, at least informally, what a ground field is. Chi