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I'm looking for a basic motivation for this topic. Meaning, why would one want to study the function field of a curve. The reason I usually give to this question is that one understands a space by understanding the functions that can be defined on it. Also, this object categorically classifies curves up to birational equivalence.

However, to motivate a younger student who doesn't know anything about birational geometry nor indeed has yet developed the intuition about understanding geometry through functions, these reasons can seem arbitrary. Does anyone know of a way or analogy to see why one would come to study function fields? In particular, of curves?

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This seems backwards. If your students don't understand how to see geometry through functions, this is a perfect opportunity to teach them! It is easy to explain through examples how a morphism of function fields gives rise to a morphism of projective curves in the opposite direction (or a birational map of affine curves) and you can talk about this as a geometric way to think about Galois theory. –  Qiaochu Yuan Feb 28 '11 at 18:47
    
That is basically what I plan to do if I don't find another motivation, though they haven't had any Galois theory yet. I was just wondering if there was something more tangible for them. –  lmiller Feb 28 '11 at 19:17
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Why not strictly starting from algebra? In field theory one studies field extensions $F/K$ because every field is an extension of its prime field. Field extensions can be algebraic or transcendental. In the first case a detailled theory exists. In the second case the simplest class consists of finitely generated extensions of transcendence degree one. To describe their structure one necessarily has to look at a set of generators and the relations among them. These relations yield a curve over the base field. –  Hagen Mar 1 '11 at 11:24
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