2
$\begingroup$

I'm using Stewart's Galois Theory book, which states that a field extension is a pair L:K such that there exists a monomorphism between K and L, and both of them are subfields of $\mathbb{C}$.

I have a problem with considering the set of all rational expressions over K denoted K(t) as a subfield of $\mathbb{C}$, because strictly speaking it is a set of expressions. Then K(t):K would not be a field extension.

Thank you in advance for your help.

$\endgroup$
  • 1
    $\begingroup$ Are you he demands both to be subfields of $\mathbb C$? I'll,bet you mis-read something since that is certainly not a demand. $\endgroup$ – Ittay Weiss Dec 31 '16 at 17:30
  • 1
    $\begingroup$ On the other hand, for example $\Bbb Q(t)$ can be viewed as subfield of $\Bbb C$ by mapping $t\mapsto \pi$ or any other transcendental $\endgroup$ – Hagen von Eitzen Dec 31 '16 at 17:32
  • $\begingroup$ @IttayWeiss Yes I'm sure, I think it's a typo, but I'm glad it is not a demand, thank you very much. $\endgroup$ – Santiago Estupiñán Dec 31 '16 at 17:41
  • $\begingroup$ @HagenvonEitzen It is a useful way of visualize it as a subfield. Thank you very much $\endgroup$ – Santiago Estupiñán Dec 31 '16 at 17:42
2
$\begingroup$

The notion of field extension is indeed more general and includes just any pair of fields $K$, $L$ with a (fixed) monomorphism $K\rightarrow L$.

If $K$ is any field and $L$ an extension of degree of transcendence not greater than the degree of transcendence of $K\subset \overline{K}$ (an overline denotes the algebraic closure of the field), the field $L$ can always be seen as a subfield of $\overline K$.

This fact justifies the so called Lefschetz principle which somewhat vaguely and in down-to-earth terms says that to do algebraic geometry in characteristic $0$ it is sufficient to work in the complex field $\Bbb C$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.