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

prime field
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15A99 no label found12F99 no label found12E99 no label found12E20 no label found


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.


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

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

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