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Definition: a field $L\supseteq K$ is called a function field over $K$ if the extension $L|K$ is finitely generated, regular and of transcendence degree $1$.


In the book "Topics in the theory of algebraic function fields", at page 25 you can find the following sentence:

In the number field case, there exist archimedean absolute values. In our case, the function field case, all absolute values are nonarchimedean.

Why is this true? In practice a function field $L$ doesn't have infinite primes.

Edit: In characteristic $p$ this is not very hard to see, but the problems arise in characteristic $0$.

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This certainly isn't true over fields of characteristic $0$. Any embedding of $L$ in $\mathbb{C}$ gives an archimedean absolute value on $L$ by restriction, and such an embedding exists as long as $L$ has characteristic $0$ and cardinality $\leq 2^{\aleph_0}$.

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