Function Field (scheme Theory) - Further Issues

Further Issues

Once KX is defined, it is possible to study properties of X which depend only on KX. This is the subject of birational geometry.

If X is an algebraic variety over a field k, then over each open set U we have a field extension KX(U) of k. The dimension of U will be equal to the transcendence degree of this field extension. All finite transcendence degree field extensions of k correspond to the rational function field of some variety.

In the particular case of an algebraic curve C, that is, dimension 1, it follows that any two non-constant functions F and G on C satisfy a polynomial equation P(F,G) = 0.

Read more about this topic:  Function Field (scheme Theory)

Famous quotes containing the word issues:

    How to attain sufficient clarity of thought to meet the terrifying issues now facing us, before it is too late, is ... important. Of one thing I feel reasonably sure: we can’t stop to discuss whether the table has or hasn’t legs when the house is burning down over our heads. Nor do the classics per se seem to furnish the kind of education which fits people to cope with a fast-changing civilization.
    Mary Barnett Gilson (1877–?)

    The “universal moments” of child rearing are in fact nothing less than a confrontation with the most basic problems of living in society: a facing through one’s children of all the conflicts inherent in human relationships, a clarification of issues that were unresolved in one’s own growing up. The experience of child rearing not only can strengthen one as an individual but also presents the opportunity to shape human relationships of the future.
    Elaine Heffner (20th century)