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60 Appendix to Chapter 2

+
0 ≤ ord ( f ) = ord ( f 1 ) +· · · + ord ( f r ).
+
+
p p p
+
We can assume that ord ( f i ) ≥ 0 for all i and that p is irreducible in R. This means
p
that each f i is an irreducible in R[x].
Let f be an irreducible in R[x] with f (x) dividing a(x)b(x) in R[x]. If deg( f ) =
0, then f is an irreducible in R and
+
+
+
0 < ord (a(x)b(x)) = ord (a(x)) + ord (b(x)).
f f f
+
+
Hence f divides a(x) when ord (a(x)) > 0or b(x) when ord (b(x)) > 0. If
f f
+
deg( f )> 0, then ord ( f ) = 0 for every irreducible p ∈ R and f is irreducible in
p
F[x] by the argument in the previous paragraph. Since F[x] is factorial, f (x) divides
+
a(x) or b(x) in F[x]and so a(x) = f (x)q(x) for example. Then 0 ≤ ord (a(x)) =
p
+
+
+
ord ( f (x))+ord (q(x)) = ord (q(x)) and q(x) ∈ R[x]. Hence f (x) divides a(x)
p
p
p
in R[x] with quotient q(x). This proves the theorem.
§3. Remarks on Valuations and Algebraic Curves
Let k be an algebraically closed field, and let V denote the set of valuations up to
equivalence on the field k(x) of rational functions in one variable over k which are
trivial on k. We map P 1 (k) → V by
(1, a) in P 1 (k) → v (1,a) = ord x−a for k[x] ⊂ k(x),
∞= (0, 1) in P 1 (k) → v (0,1) = ord 1/x for k[1/x] ⊂ k(x).
Then the function P 1 (k) → V given by P → v p , is a bijection. For it is clearly
an injection, and to show that each valuation v is of the form v p , consider the ideal
M v ∩ k[x]in k[x]if x ∈ R v or the ideal M v ∩ k[1/x]if x ∈ R v . The remainder of
the argument is left to the reader.
(3.1) Remark. Let C f be an irreducible plane curve defined over k. Then the inte-
gral domains

k[x, y] k[w, y] k[w, x]
, ,
f (1, x, y) f (w, 1, y) f (w, x, 1)
all have naturally isomorphic fields of fractions, denoted k(C f ), and called the func-
tion field of the curve. The subrings k[w], k[x], and k[y] all inject into k(C f ) except
when f is w, x, or y, respectively. Thus the field k(C f ) is a finite extension of
k(w), k(x), or k(y). The main assertion is: there is a natural map from C f (k) to val-
uations on k(C f ) trivial over k up to equivalence which is a bijection when C f is
nonsingular and k is algebraically closed. This map extends P 1 (k) → V considered
above in the sense that for k(P 1 ) = k(t) ⊂ k(C f ) the points of C f mapping onto
P ∈ P 1 under t : C f → P 1 correspond to the extensions to k(C f ) of the valuation
v p on k(P 1 ).