What is the difference between the algebraic function fields and the fields itself

Question

I'm studying this book and I don't understand exactly what's the difference between the algebraic function field $F/K$ and $F$ itself.

Thanks

Answer 1

The notation $F / K$ is just a traditional notation that indicates that $F$ is an extension of $K$, i.e. that $F$ contains $K$. So, for now, you can regard $F / K$ as the same thing as $F$, with the notation reminding you that $F$ contains the subfield $K$.

When you start to study maps between fields, then the distinction takes on more significance. An automorphism of a field $F$ is a bijective map $f : F \rightarrow F$ such that $f(x+y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$. An automorphism of $F/K$ is an automorphism $f$ of $F$ that acts as the identity on $K$, i.e. $f(x) = x$ for all $x \in K$. Thus all automorphisms of $F/K$ are automorphisms of $F$, but not vice versa.

As an example, there are infinitely many automorphisms of $\mathbb{C}$ (though all but the two mentioned below are "wild" and hard to describe), but there are only two automorphisms of $\mathbb{C} / \mathbb{R}$, namely the identity map and the complex conjugation map.