We see that the degree of an algebraic function field is not a well-defined notion.
The algebraic function fields over k form a category; the morphisms from function field K to L are the ring homomorphisms f : K → L with f(a) = a for all a in k. All these morphisms are injective.
(But note that non-isomorphic varieties may have the same function field!)
Assigning to each variety its function field yields a duality (contravariant equivalence) between the category of varieties over k (with dominant rational maps as morphisms) and the category of algebraic function fields over k. (The varieties considered here are to be taken in the scheme sense; they need not have any k-rational points, like the curve X2 + Y2 + 1 = 0 defined over the reals, that is with k = R.) The case n = 1 (irreducible algebraic curves in the scheme sense) is especially important, since every function field of one variable over k arises as the function field of a uniquely defined regular (i.e. non-singular) projective irreducible algebraic curve over k. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves (with dominant regular maps as morphisms) and the category of function fields of one variable over k. The field M(X) of meromorphic functions defined on a connected Riemann surface X is a function field of one variable over the complex numbers C. In fact, M yields a duality (contravariant equivalence) between the category of compact connected Riemann surfaces (with non-constant holomorphic maps as morphisms) and function fields of one variable over C. A similar correspondence exists between compact connected Klein surfaces and function fields in one variable over R. The function field analogy states that almost all theorems on number fields have a counterpart on function fields of one variable over a finite field, and these counterparts are frequently easier to prove.
(For example, see Analogue for irreducible polynomials over a finite field.)
Given an algebraic function field K/k of one variable, we define the notion of a valuation ring of K/k: this is a subring O of K that contains k and is different from k and K, and such that for any x in K we have x ∈ O or x -1 ∈ O.
A discrete valuation of K/k is a surjective function v : K → Z∪{∞} such that v(x) = ∞ iff x = 0, v(xy) = v(x) + v(y) and v(x + y) ≥ min(v(x),v(y)) for all x, y ∈ K, and v(a) = 0 for all a ∈ k \ {0}.
These sets can be given a natural topological structure: the Zariski–Riemann space of K/k.