4 – The Riemann Surface of an Algebraic Function
305
be primitive if c(f ) = 1. Show that if f and g are primitives, then so is
fg (“ Gauss’ Lemma ”). (c) Show that c(fg) = c(f )c(g) for all f, g ∈ Z[X].
(d) Show that the preceding arguments and results continue to hold if Z is
replaced by the ring A = k[X] of polynomials in one variable with coefficients
in the field k [replace “ prime ” with “ irreducible ”] and more generally by
an arbitrary principal ring A. (e) Let P (X, Y ) be an irreducible polynomial
with coefficients in a field k. Show that as a polynomial in Y with coefficients
in the field K = k(X) of rational fractions, P is still irreducible. In other
words: if P has a non-trivial divisor in K[X] = k(X)[Y ], then it also has one
in k[X, Y ].
(v) Meromorphic functions on ˆ
X.
Theorem 8. All meromorphic functions on ˆ
X are rational functions of z
and ζ.
We will assume a result from the general theory of commutative field
extensions though it is not hard to show. Otherwise the proof given will be
complete.
Let ϕ be a meromorphic function on ˆ
X. It has finitely many poles projecting onto points a i ∈ ˆ
C . Let B ϕ be the open set obtained by removing
from B the points a i belonging to it. If, for some z ∈ B ϕ , the n (distinct)
points of X over z are numbered by (z, ζ k ), then the expression
(T − ϕ (z, ζ k )) = T
n + c 1 (z)T
n
− 1 + . . . + c n (z)
(4.13)
is well-defined. Its coefficients, i.e. the elementary symmetric functions of
ϕ(z, ζ k ) up to sign, are defined on all of B ϕ . These are holomorphic functions
on B ϕ because, over a sufficient small disc D ⊂ B ϕ centered at a, the covering
space X decomposes into “ discs ” D(α k ) on which ϕ is a holomorphic function
of the local uniformizer q, and so of z, whence the result. As, on the other
hand, the function ϕ is O(q
−N ) in the neighbourhood of any point that does
not project onto B ϕ , where q is the local uniformizer at this point, the c i (z)
have at most poles at the point of ˆ
C − B ϕ , hence are rational functions of z.
If M denotes the field of meromorphic functions on ˆ
X and if every rational
function f (z) on ˆ
C is identified with the function (z, ζ) → f (z) on ˆ
X, then
it follows that every ϕ ∈ M is algebraic of degree ≤ n on K = C(z).
On the other hand, M contains the field L of rational functions of z and
ζ . As seen at the end of the previous section (iv), the polynomial P (X, Y )
is irreducible as a polynomial in Y with coefficients in K. The following very
simple result then shows L has dimension n over K :
Lemma 2. Let M be a commutative field, K a subfield of M and ζan element
of M satisfying an irreducible algebraic equation of degree n over K. Then
the subfield L of M generated by K and ζ has dimension n over K and admits
1, ζ, . . . , ζ
n−1 as a basis over K.
305
be primitive if c(f ) = 1. Show that if f and g are primitives, then so is
fg (“ Gauss’ Lemma ”). (c) Show that c(fg) = c(f )c(g) for all f, g ∈ Z[X].
(d) Show that the preceding arguments and results continue to hold if Z is
replaced by the ring A = k[X] of polynomials in one variable with coefficients
in the field k [replace “ prime ” with “ irreducible ”] and more generally by
an arbitrary principal ring A. (e) Let P (X, Y ) be an irreducible polynomial
with coefficients in a field k. Show that as a polynomial in Y with coefficients
in the field K = k(X) of rational fractions, P is still irreducible. In other
words: if P has a non-trivial divisor in K[X] = k(X)[Y ], then it also has one
in k[X, Y ].
(v) Meromorphic functions on ˆ
X.
Theorem 8. All meromorphic functions on ˆ
X are rational functions of z
and ζ.
We will assume a result from the general theory of commutative field
extensions though it is not hard to show. Otherwise the proof given will be
complete.
Let ϕ be a meromorphic function on ˆ
X. It has finitely many poles projecting onto points a i ∈ ˆ
C . Let B ϕ be the open set obtained by removing
from B the points a i belonging to it. If, for some z ∈ B ϕ , the n (distinct)
points of X over z are numbered by (z, ζ k ), then the expression
(T − ϕ (z, ζ k )) = T
n + c 1 (z)T
n
− 1 + . . . + c n (z)
(4.13)
is well-defined. Its coefficients, i.e. the elementary symmetric functions of
ϕ(z, ζ k ) up to sign, are defined on all of B ϕ . These are holomorphic functions
on B ϕ because, over a sufficient small disc D ⊂ B ϕ centered at a, the covering
space X decomposes into “ discs ” D(α k ) on which ϕ is a holomorphic function
of the local uniformizer q, and so of z, whence the result. As, on the other
hand, the function ϕ is O(q
−N ) in the neighbourhood of any point that does
not project onto B ϕ , where q is the local uniformizer at this point, the c i (z)
have at most poles at the point of ˆ
C − B ϕ , hence are rational functions of z.
If M denotes the field of meromorphic functions on ˆ
X and if every rational
function f (z) on ˆ
C is identified with the function (z, ζ) → f (z) on ˆ
X, then
it follows that every ϕ ∈ M is algebraic of degree ≤ n on K = C(z).
On the other hand, M contains the field L of rational functions of z and
ζ . As seen at the end of the previous section (iv), the polynomial P (X, Y )
is irreducible as a polynomial in Y with coefficients in K. The following very
simple result then shows L has dimension n over K :
Lemma 2. Let M be a commutative field, K a subfield of M and ζan element
of M satisfying an irreducible algebraic equation of degree n over K. Then
the subfield L of M generated by K and ζ has dimension n over K and admits
1, ζ, . . . , ζ
n−1 as a basis over K.