Elementary proof of finitely generateness of a subring of k[t]

Marek Karaś

IMUJ Preprint 1999/05

Abstract

In this note we present elementary proof of known fact that any subring R ⊂ k[t] which contains a field k, where k[t] is the ring of polynomials in one variable, is finitely generated k−algebra. In our proof only Chinese Remainder Theorem is used.

1 Introductions

By ℕ, ℤ we denote sets of natural and integer numbers, respectively. We assume that 0 ∈ ℕ. By k, k[t], k[X_{1}, …, X_{n}] and k(X_{1}, …, X_{n}) we denote a field, ring of polynomials in one variable, ring of polynomials in n variables and field of rational functions in n variables, respectively.

The fourteenth problem of Hilbert is the following question:

1 Let L ⊂ k(X_{1}, …, X_{n}) be arbitrary subfield such that k ⊂ L. Whether L ∩ k[X_{1}, …, X_{n}] is a finitely generated k−algebra.

More general quastion is the following:

2 Let R ⊂ k[X_{1}, …, X_{n}] be arbitrary subring such that k ⊂ R. Whether R is a finitely generated k−algebra.

For n > 1 answer to the second quastion is, in general, negative. For some more details see e.g. [1]. For n = 1 answer is given by the following known

Theorem 1 If R is a subring of k[t] such that k ⊂ R then R is a finitely generated k−algebra.

In this note an elementary proof of the above Theorem is given. In our proof we use only Chinese Remainder Theorem and linear algebra.

2 Numerical Lemma

If k, l ∈ ℤ and k = l · m for some m ∈ ℤ then we will write l|k. If k, l, m ∈ ℤ and m|(k − l) then we will write k ≡ l (mod m). For any m_{1}, …, m_{n} ∈ ℤ by GCD(m_{1}, …, m_{n}) we denote the greatest common divisor of numbers m_{1}, …, m_{n} and assume that GCD(m_{1}, …, m_{n}) ∈ ℕ.

Lemma 2 Let n ∈ ℕ, m_{1}, …, m_{n} ∈ ℕ and GCD(m_{1}, …, m_{n}) = d. There exists l_{0} ∈ ℕ such that if l ≥ l_{0} and d|l then there exist l_{1}, …, l_{n} ∈ ℕ such that

l = l_{1}m_{1} + … + l_{n}m_{n}.

Proof: Let

ε_{j} = jd for j = 0, 1, …, u = \frac{m_{1}}{d} − 1.

By Chinese Remainder Theorem there exist l_{j,1}, …, l_{j,n} ∈ ℤ, j = 0, 1, …, u, such that

ε_{j} = l_{j,1}m_{1} + … + l_{j,n}m_{n}.

Let

I_{j} = {i ∈ {1, …, n} : l_{j,i} < 0}, j = 0, 1, …, u.

For

ε̃_{j} = ε_{j} + Σ_{i∈I_{j}}(|l_{j,i}|m_{1})m_{i},

we have

ε̃_{j} ≡ ε_{j}(mod m_{1})

and

ε̃_{j} = Σ_{i∉I_{j}}l_{j,i}m_{i} + Σ_{i∈I_{j}}(l_{j,i} + |l_{j,i}|m_{1})m_{i}, for j = 0, 1, …, u.

Thus ε̃_{0}, …, ε̃_{u} can be written as linear combinations of m_{1}, …, m_{n} with coefficients from ℕ.

For any l ≥ l_{0} = max{ε̃_{0}, …, ε̃_{u}} such that d|l there exists j ∈ {0, 1, …, u} such that

l ≡ ε_{j} (mod m_{1}).

Thus

l = ε̃_{j} + l̃m_{1}

for some l̃∈ ℕ, and so l can be written as a linear combination of m_{1}, …, m_{n} with coefficients from ℕ. □

3 Proof of the Theorem

We can assume that k ≠ R. Let

R_{p} = {f ∈ R : deg f ≤ p} for p ∈ ℕ.

The set R_{p}, for any p ∈ ℕ, is a finite dimensional linear space over k.

Let

M = {deg f : f ∈ R\k}.

We have

M = {m_{1}, m_{2}, …},

where m_{1}, m_{2}, … ∈ ℕ. Let

μ_{1} = m_{1}, μ_{2} = GCD(m_{1}, m_{2}), …, μ_{n} = GCD(m_{1}, …, m_{n}), …

We have

μ_{n} ≥ μ_{n+1} and μ_{n} ∈ ℕ for n ∈ ℕ.

Thus there exists n_{0} ∈ ℕ such that

μ_{n_{0}} = μ_{n_{0}+k} for k ∈ ℕ.

We put μ = μ_{n_{0}}.

Let f_{1}, …, f_{n_{0}} ∈ R be such that

deg f_{i} = m_{i} for i = 1, …, n_{0}

and g_{1}, …, g_{s} be any basis of the k−linear space R_{l_{0}}, where l_{0} is from Lemma 2 (for numbers m_{1}, …, m_{n_{0}}).

Let f ∈ R be arbitrary. If deg f ≤ l_{0} then

f = a_{1}g_{1} + … + a_{s}g_{s}

for some a_{1}, …, a_{s} ∈ k. If deg f > l_{0} then μ|deg f and by Lemma 2 there exist l_{1}, …, l_{n_{0}} ∈ ℕ such that

deg f = l_{1}deg f_{1} + … + l_{n_{0}}deg f_{n_{0}}.

Then there exists a ∈ k such that

deg (f − af_{1}^{l_{1}} · . . . · f_{n_{0}}^{l_{n_{0}}}) < deg f.

By repeating of the above operation we can construct f̃ ∈ k[f_{1}, …, f_{n_{0}}] such that

deg (f − f̃) ≤ l_{0}.

The above inequality means that f − f̃ ∈ R_{l_{0}}, and finally we get that

f ∈ k[g_{1}, …, g_{s}, f_{1}, …, f_{n_{0}}].

□

References

[1] A. Nowicki, Przykład Freudenburga do czternastego problemu Hilberta, Materiały na XX konf. szkol. z Geom. Anal. i Alg. Zespolonej, Łódż 1999