Hello guys i am trying to solve excerise 2.7 page 14 from Gathmann notes https://agag-gathmann.math.rptu.de/class/curves-2023/curves-2023-c2.pdf
Definition
About (a) : Stuck here.Not sure how to use the curves not having common component at the origin.
About (b) : If the base field was the complex numbers i could use (a) to show the power series of a an element in K(x,y) has finite terms in the quotient,so i could get a polynomial representative that way.But since the field is random not sure how..maybe use geometric series?
About (c): i can show using (b) and the form of the representative that the quotient is spanned by finite amount of elements,so as a K-vector space it should have finite dimension.
Any hints that could point me to the right direction (especially about (a)) would be welcome.