Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 1, 57-66.

Freeness, Linear Disjointness, and Implicitization – a Classical Approach

Rainer Steinwandt^{1} Jörn Müller-Quade

Institut für Algorithmen und Kognitive Systeme Arbeitsgruppe Computeralgebra, Fakultät für Informatik Universität Karlsruhe, Germany

Abstract. A method for deciding linear disjointness and freeness of finitely generated extension fields is given. The techniques used are based on a classical description of linear disjointness and on the Chow form. The required Gröbner basis techniques do not depend on tag variables. Finally, the solution obtained can also be used to solve the implicitization problem without involving tag variables.

1. Introduction

Let k(w) := k(w_{1}, …, w_{l}) be a field finitely generated over a ground field k, k(x) := k(x_{1}, …, x_{m}) and k(y) := k(y_{1}, …, y_{n}) subfields of k(w), k(z) := k(z_{1}, …, z_{o}) an intermediate field of k and k(x) ∩ k(y). In [9] a method for solving the following problems is given:

1. Decide if k(x) and k(y) are linearly disjoint over k(z).

2. Decide if k(x) and k(y) have a subfield k(z′) in common such that k(x) and k(y) are linearly disjoint over k(z′) (where z′_{1}, …, z′_{o′} ∈ k(w)).

3. For k(x) and k(y) being linearly disjoint (necessarily over their intersection (see [9, Prop. 15])) construct generators for k(x) ∩ k(y) over k.

4. Decide if k(x) is free from k(y) over k(z).

1grant DFG - GRK 209/3-98 “Beherrschbarkeit komplexer Systeme”

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

The present paper gives an alternate solution to the above problems which is based on the classical descriptions of linear disjointness and freeness as given in [18]. By means of the Chow form we obtain a solution which in contrast to the method described in [9] does not make use of so-called tag variables. Gröbner basis computations involving many variables tend to be costly. Therefore we do without these additional variables and make use of extensions of the ground field instead. The idea of extending the ground field in order to avoid additional variables also occurs in the context of the inversion of birational maps ([13]) and the functional decomposition of rational mappings ([10]), for instance.

In Section 5 we describe how the same technique can also be applied to solve the implicitization problem in computer aided geometric design without making use of tag variables.

In a short annotation we point out that in a sense the use of Gröbner basis techniques for deciding linear disjointness was suggested in [18] already. Therefore it might be appropriate to take Weil’s work into account when dealing with the history of Gröbner bases.

2. Freeness, linear disjointness, and the Chow form

The key to the constructive solution of the above problems is the ideal^{1}

𝔓_{(y)/k(x)} := {p ∈ k(x)[Z_{1}, …, Z_{n}] : p(y_{1}, …, y_{n}) = 0}.

To illustrate its importance we remind of the following characterization of linear disjointness:

Recall 1. [18, Ch. I, §6, Theorem 3] Let k′ ≤ k(x). Then k(x) and k′(y) are linearly disjoint over k′ if and only if 𝔓_{(y)/k(x)} has a basis in k′[Z] := k′[Z_{1}, …, Z_{n}].

In order to use this result to obtain a solution for the first of the above stated problems we also need

Recall 2. [18, Ch. I, §7, Lemma 2] Let ℑ ⊴ k[Z] be an ideal. Then, of all the subfields k′ of k such that ℑ has a basis with coefficients in k there is a smallest field k_{0}, contained in all the others.

In the sequel this field k_{0} is referred to as the minimal field of definition of ℑ. Using this terminology we immediately obtain the following three conceptual steps for solving the first of the above problems by applying Recall 1 to k′ = k(z):

1. Construct a basis of 𝔓_{(y)/k(x)}.

2. Derive a finite generating set of 𝔓_{(y)/k(x)} with all coefficients being contained in the minimal field of definition of 𝔓_{(y)/k(x)}. Denote the set of coefficients in this basis by B.

3. Now k(x) and k(y) are linearly disjoint over k(z) if and only if B ⊆ k(z) (cf. [10] for

deciding field membership without tag variables). The second and third problem can be solved in the same manner: To check whether there is a common subfield k(z′) of k(x) and k(y) with k(x) and k(y) being linearly disjoint over k(z′) we simply have to verify whether the above set B (which by definition is a subset of k(x)) is contained in k(y). Similarly, for the third problem we apply Recall 1 to k′ := k(x) ∩ k(y): If the fields k(x) and k(y) happen to be linearly disjoint, the set B is necessarily the required generating set of k(x) ∩ k(y) over k.

1In [9] this ideal is denoted by J_{k(y)/k(x)}. However, as the definition depends on the generating set y, we adopt the notation of [18].

Before dealing with the above three steps in detail we first consider the simpler problem of deciding freeness, as the technique used here will also prove useful when dealing with linear disjointness. The key for deciding freeness constructively is

Recall 3. [18, Ch. I, §2, Proposition 2] k(x) is free from k(y) over k(z) if and only if transdeg(k(x, y)/k(x)) = transdeg(k(y, z)/k(z)).

To determine the required transcendence degrees we can use the trivial identitity

transdeg(k(x, y)/k(x)) = transdeg(k(w)/k(x)) − transdeg(k(w)/k(x, y))

(and analogously transdeg(k(y, z)/k(z)) = transdeg(k(w)/k(z)) − transdeg(k(w)/k(y, z))).

For computing transdeg(k(w)/k(x)) resp. transdeg(k(w)/k(x, y)) [10, Lemma 2] suggests to determine the dimension of the prime ideal 𝔓_{(w)/k(x)} resp. 𝔓_{(w)/k(x,y)}. This can be done by means of a Gröbner basis computation, for instance ([8, Theorem 1]). Using the method in [10] a basis of 𝔓_{(w)/k(x)} can be found without introducing tag variables; an approach to determine the transcendence degree by means of tag variables is discussed in [16] and [6].

In summary we have

Lemma 4. Procedure 1 decides for finite subsets x, y, z ⊆ k(w) whether k(x) is free from k(y) over k(z) without introducing tag variables.

Procedure 1

In: x, y, z ⊆ k(w) Out: true, if k(x) is free from k(y) over k(z)

false, otherwise

begin G_{1} ← any Gröbner basis of 𝔓_{(w)/k(x)} (see [10]). G_{2} ← any Gröbner basis of 𝔓_{(w)/k(x,y)} (see [10]). G_{3} ← any Gröbner basis of 𝔓_{(w)/k(z)} (see [10]). G_{4} ← any Gröbner basis of 𝔓_{(w)/k(y,z)} (see [10]). Derive dim(𝔓_{(w)/k(x)}), dim(𝔓_{(w)/k(x,y)}), dim(𝔓_{(w)/k(z)}), and dim(𝔓_{(w)/k(y,z)}) from

G_{1}, G_{2}, G_{3}, and G_{4} via [8, Theorem 1].

if dim(𝔓_{(w)/k(x)}) − dim(𝔓_{(w)/k(x,y)}) = dim(𝔓_{(w)/k(z)}) − dim(𝔓_{(w)/k(y,z)})

then return true else return false fi end Deciding linear disjointness is a bit more involved, as we do not only need the dimension but also a generating set of 𝔓_{(y)/k(x)}. To reach this goal without using tag variables we can apply a classical tool from algebraic geometry, namely, the Chow form:

Denote by t := dim(𝔓_{(x)/k}) = transdeg(k(x_{1}, …, x_{m})/k) the transcendence degree of the extension k(x_{1}, …, x_{m})/k, and let {u_{ij} : i = 1, …, t + 1, j = 1, …, m} be algebraically independent over k(x). Furthermore, we write F_{(x)/k}(u; Z) ∈ k[u_{11}, …, u_{t+1m}, Z_{1}, …, Z_{t+1}] \ {0} for the up to a constant factor in k^{×} unique irreducible polynomial with

F_{(x)/k}(u; ∑_{j=1}^{m} u_{1j}x_{j}, …, ∑_{j=1}^{m} u_{t+1j}x_{j}) = 0.

Then following [15] we refer to F_{(x)/k} as the Chow form of 𝔓_{(x)/k}. In particular, the number of indeterminates of the form Z_{i} occurring in F_{(x)/k} equals transdeg(k(x)/k). Note that if F_{(x)/k} did not contain all of Z_{1}, …, Z_{t+1} there would be an algebraic dependence over k(u) among t sums of the form ∑ u_{ij}x_{j}. This would contradict the equality t = transdeg(k(x)/k) = transdeg(k(u)(x)/k(u)).

Defining polynomials of the variety V(𝔓_{(x)/k}) in affine m-space over an algebraic closure of k can be derived from F_{(x)/k} by means of [17, §5 B.]: Introduce new variables X_{1}, …, X_{m}, and for i = 1, …, t + 1 replace Z_{i} by ∑_{j=1}^{m} u_{ij}X_{j} in F_{(x)/k}(u; Z). Then the coefficients of

F_{(x)/k}(u; ∑_{j=1}^{m} u_{1j}X_{j}, …, ∑_{j=1}^{m} u_{t+1j}X_{j}) ∈ k[X][u]

form an ideal 𝔄_{(x)/k} ⊴ k[X] having V(𝔓_{(x)/k}) as associated locus. Unfortunately 𝔄_{(x)/k} is not necessarily prime, and can even have embedded primes (see [15] for an example). However, 𝔄_{(x)/k} has a unique isolated primary component whose associated prime equals 𝔓_{(x)/k}. In particular, we can determine generators for 𝔓_{(x)/k} by computing the associated primes of 𝔄_{(x)/k} (see [14], [4] and the references given there), followed by choosing the associated prime of maximal dimension (again, the dimension of the prime ideals can be determined by means of [8, Theorem 1]).

Note that 𝔓_{(x)/k} is in particular the radical of the equidimensional hull of 𝔄_{(x)/k} (resp. for k(x)/k separable equal to the equidimensional hull of 𝔄_{(x)/k}, as in this case the isolated primary component is prime (see [7, Theorem 5])). Therefore also the algorithms in [4] for determining the (radical of the) equidimensional hull of an ideal can be applied here.

Having in mind the three conceptual steps given above we are left to find procedures for • computing the Chow form F_{(x)/k} and

• determining the minimal field of definition k_{0} of an ideal ℑ ⊴ k[Z].

3. Computing the Chow form and the minimal field of definition of an ideal

If the transcendence degree t of k(x)/k is known then finding F_{(x)/k} reduces to computing the minimal polynomial of ∑_{j=1}^{m} u_{t+1j}x_{j} over k(x, ∑ u_{1j}x_{j}, …, ∑ u_{tj}x_{j}). In our concrete situation we have to find the transcendence degree of an extension of the form k(x, y)/k(x) where x and y are rational functions in the generators w of k(w). For this we can proceed as in the above procedure for deciding linear disjointness, namely, compute

transdeg(k(x, y)/k(x)) = dim(𝔓_{(w)/k(x)}) − dim(𝔓_{(w)/k(x,y)})

via [10] and [8, Theorem 1].

Finally, to find the required minimal polynomial itself—again without tag variables—we can reuse the Gröbner basis of 𝔓_{(w)/k(x)}, which has proven useful in determining the transcendence degree, by applying [11, Algorithm 3.2]. The latter procedure actually was designed for computing minimal polynomials over intermediate fields of a purely transcendental extension k(w)/k; this restriction is not required in the proof of its correctness, however, and it is sufficient to have a Gröbner basis of the ideal 𝔓_{(w)/k(x)} to apply this algorithm.

For finding the minimal field of definition k_{0} of an ideal ℑ ⊴ k[Z] in [9] and [12] the following approach is suggested: Determine a reduced Gröbner basis G_{ℑ} of ℑ. Then k_{0} is the field which over the prime field of k is generated by the coefficients of the elements of G_{ℑ}. The correctness of this method is verified easily: G_{ℑ} is unique and —via Buchberger’s algorithm—can be derived from any finite basis of ℑ without extending the field generated (over the prime field) by the coefficients of the polynomials in this initial generating set.

At first glance using a reduced Gröbner basis for determining the field of definition may seem a bit strange. But in fact, already the proof of the existence of the minimal field of definition of an ideal given in [18] makes use of a finite canonical generating set. To illustrate the connection between these (Weil) bases and Gröbner bases we restate part of Weil’s proof of Recall 2.

To derive this result Weil looks at the set of terms (=monic monomials) T(Z) in the variables Z. Using some linear order ⪯ we may take T(Z) for a w. r. t. ⪯ increasing sequence (t_{i})_{i=0}^{∞}. Omitting all terms t_{i} from T(Z) with t_{i} − ∑_{j=0}^{i−1} α_{j}t_{j} ∈ ℑ for some α_{j} ∈ k we obtain a subsequence (t_{i_{λ}})_{λ=0}^{∞}. Now define the sequence of polynomials (p_{i})_{i=0}^{∞} via

p_{i} := \{ \begin{matrix}0, & if i = i_{λ} for some λ ∈ ℕ_{0}, \\ t_{i} − ∑_{i_{λ}<i} α_{λ}t_{i_{λ}} with α_{λ} ∈ k such that t_{i} − ∑_{i_{λ}<i} α_{λ}t_{i_{λ}} ∈ ℑ, & otherwise. \end{matrix}

Using Hilbert’s basis theorem Weil concludes that for some r ∈ ℕ_{0} the polynomials p_{0}, …, p_{r} form a basis of ℑ. At this conclusion it is implicitly used that ⪯ is of type ω, otherwise the required integer r need not exist (think of ℑ = ⟨Z_{1}, Z_{2}⟩ ⊴ k[Z_{1}, Z_{2}] and ⪯ a lexicographical term order). So after fixing a linear order of type ω and choosing r minimal with p_{0}, …, p_{r} being a basis of ℑ the set W_{⪯}(ℑ) := ⋃_{i=0}^{r}{p_{i}}\{0} forms a canonical generating set of ℑ. Weil proves that the coefficients of the polynomials in this set over the prime field of k generate the required field of definition k_{0}.

Observation 5. In general this canonical (Weil) basis is not a Gröbner basis w. r. t. ⪯.

Proof. Let ℑ = ⟨Z_{1}^{2} − Z_{2}, Z_{1}Z_{2}^{2} − 1⟩ ⊴ ℚ[Z_{1}, Z_{2}] and denote by ⪯ the graded reverse lexicographic term order with Z_{1} ≺ Z_{2}. Then the sequence of terms (t_{i})_{i=0}^{∞} starts with

(t_{0}, t_{1}, t_{2}, …) = (1, Z_{1}, Z_{2}, Z_{1}^{2}, Z_{1}Z_{2}, Z_{2}^{2}, Z_{1}^{3}, Z_{1}^{2}Z_{2}, Z_{1}Z_{2}^{2}, Z_{2}^{3}, …) ,

and we obtain

(p_{0}, p_{1}, p_{2}, …) = (0, 0, 0, Z_{1}^{2} − Z_{2}, 0, 0, Z_{1}^{3} − Z_{1}Z_{2}, Z_{1}^{2}Z_{2} − Z_{2}^{2}, Z_{1}Z_{2}^{2} − 1, Z_{2}^{3} − Z_{1}) .

So the canonical (Weil) basis computes to

W_{⪯}(ℑ) = {p_{3}, p_{4}, p_{5}, p_{6}} = {Z_{1}^{2} − Z_{2}, Z_{1}^{3} − Z_{1}Z_{2}, Z_{1}^{2}Z_{2} − Z_{2}^{2}, Z_{1}Z_{2}^{2} − 1} .

In particular, it is not a Gröbner basis, because p_{10} = Z_{2}^{3}−Z_{1} ∈ ℑ cannot be reduced modulo W_{⪯}(ℑ). □

Note that if we add p_{10} to W_{⪯}(ℑ) where ℑ is the ideal in the above proof we obtain a Gröbner basis of ℑ. This is not by coincidence, as in fact a Weil basis can always be extended to a Gröbner basis by carrying on Weil’s procedure:

Observation 6. If ⪯ is a term order of type ω, ℑ ⊴ k[Z], and (p_{i})_{i=0}^{∞} defined as above then there exists an s ∈ ℕ_{0} such that ⋃_{i=0}^{s}{p_{i}} \ {0} is a Gröbner basis of ℑ w. r. t. ⪯.

Proof. As explained in [18, proof of Lemma 2] every polynomial in ℑ is a finite k-linear combination of some of the p_{i}, i ∈ ℕ_{0}. Hence, we may select a finite subset W ⊆ {p_{i} : i ∈ ℕ_{0}} such that all elements of the reduced Gröbner basis of ℑ w. r. t. ⪯ can be expressed as a k-linear combination of the polynomials in W. Setting s := max{i ∈ ℕ_{0} : p_{i} ∈ W } yields the desired natural number s. □

Observation 7. If W_{⪯}(ℑ) happens to be a Gröbner basis already it does neither have to be reduced nor minimal.

Proof. Denote again by ⪯ the graded reverse lexicographical term order on T(Z_{1}, Z_{2}) with Z_{1} ≺ Z_{2}. Then the sequence of terms (t_{i})_{i=0}^{∞} starts with

(t_{0}, t_{1}, t_{2}, …) = (1, Z_{1}, Z_{2}, Z_{1}^{2}, Z_{1}Z_{2}, Z_{2}^{2}, …),

and the Weil basis of ⟨Z_{1}, Z_{2}^{2}⟩ ⊴ ℚ[Z_{1}, Z_{2}] computes to {Z_{1}, Z_{1}^{2}, Z_{1}Z_{2}, Z_{2}^{2}}. In particular it is a non-minimal Gröbner basis w. r. t. ⪯. □

In summary we have

Lemma 8. Procedure 2 decides for finite subsets x, y, z ⊆ k(w) whether k(x) is linearly disjoint from k(y) over k(z) without introducing tag variables.

Procedure 2

In: x, y, z ⊆ k(w) Out: true, if k(x) and k(y) are linearly disjoint over k(z)

false, otherwise

begin Compute transdeg(k(x, y)/k(x)) (see above) Compute F_{(y)/k(x)} (using [11, Algorithm 3.2]) Derive equations for 𝔄_{(y)/k(x)} (see above) Compute the associated prime 𝔓_{(y)/k(x)} of the isolated primary component

of 𝔄_{(y)/k(x)} (using one of the methods described in [14], [4]) B ← the coefficients of a reduced Gröbner basis of 𝔓_{(y)/k(x)} w. r. t. some term order if B ⊆ k(z) (see [10]) then return true else return false fi end

Of course, to check whether there exists a common subfield k(z′) of k(x) and k(y) with k(x) and k(y) being linearly disjoint over k(z′) we simply have to verify the inclusion B ⊆ k(y) instead of B ⊆ k(z) in the if-statement. Similarly, if k(x) and k(y) are known to be linearly disjoint (necessarily over their intersection) then the set B computed above satisfies k(x) ∩ k(y) = k(B).

4. An example from invariant theory

To illustrate the above procedure we resume an example from invariant theory given in [9].

Let w_{1}, w_{2} be algebraically independent over ℂ. Then the fields ℂ(w_{1}^{50}, w_{2}^{50}, w_{1}w_{2}) and ℂ(w_{1} + w_{2}, w_{1}w_{2}) are linearly disjoint over their intersection, and (using a suitable twodimensional representation) the latter equals the field of invariants ℂ^{D_{100}} of the dihedral group D_{100} (see [9, Section 6] for details).

Applying the above ideas we can construct generators for ℂ^{D_{100}} over ℂ. For this we first determine the Chow form F_{(w_{1}^{50},w_{2}^{50},w_{1}w_{2})/ℂ(w_{1}+w_{2},w_{1}w_{2})} of 𝔓_{(w_{1}^{50},w_{2}^{50},w_{1}w_{2})/ℂ(w_{1}+w_{2},w_{1}w_{2})}: Since both ℂ(w_{1}, w_{2})/ℂ(w_{1}^{50}, w_{2}^{50}, w_{1}w_{2}) and ℂ(w_{1}, w_{2})/ℂ(w_{1} + w_{2}, w_{1}w_{2}) are algebraic we have

F_{(w_{1}^{50},w_{2}^{50},w_{1}w_{2})/ℂ(w_{1}+w_{2},w_{1}w_{2})} ∈ ℂ(w_{1} + w_{2}, w_{1}w_{2})[u_{11}, u_{12}, u_{13}; Z_{1}].

Using [11, Algorithm 3.2] to compute the minimal polynomial of u_{11}w_{1}^{50} +u_{12}w_{2}^{50} +u_{13}w_{1}w_{2} over ℂ(u)(w_{1} + w_{2}, w_{1}w_{2}) we obtain the Chow form:

Z_{1}^{2}− ((u_{11} + u_{12}) (w_{1}^{50} + w_{2}^{50}) + 2u_{13}w_{1}w_{2}) · Z_{1}
+ (u_{11}w_{2}^{50} + u_{12}w_{1}^{50} + u_{13}w_{1}w_{2}) (u_{11}w_{1}^{50} + u_{12}w_{2}^{50} + u_{13}w_{1}w_{2}).

From this we can derive defining polynomials of V(𝔓_{(w_{1}^{50},w_{2}^{50},w_{1}w_{2})/ℂ(w_{1}+w_{2},w_{1}w_{2})}) by substituting Z_{1} ← u_{11}X_{1} + u_{12}X_{2} + u_{13}X_{3} where X_{1}, X_{2}, X_{3} are new variables, followed by extracting coefficients:

X_{3}^{2} − 2w_{1}w_{2} · X_{3} + (w_{1}w_{2})^{2},
X_{1}^{2} − (w_{1}^{50} + w_{2}^{50}) · X_{1} + (w_{1}w_{2})^{50},
X_{2}^{2} − (w_{1}^{50} + w_{2}^{50}) · X_{2} + (w_{1}w_{2})^{50},
2 · X_{1}X_{3} − 2w_{1}w_{2} · X_{1} − (w_{1}^{50} + w_{2}^{50}) · X_{3} + (w_{1}^{50} + w_{2}^{50})w_{1}w_{2},
2 · X_{2}X_{3} − 2w_{1}w_{2} · X_{2} − (w_{1}^{50} + w_{2}^{50}) · X_{3} + (w_{1}^{50} + w_{2}^{50})w_{1}w_{2},
2 · X_{1}X_{2} − (w_{1}^{50} + w_{2}^{50}) · X_{1} − (w_{1}^{50} + w_{2}^{50}) · X_{2} + w_{1}^{100} + w_{2}^{100}

The ideal 𝔄_{(w_{1}^{50},w_{2}^{50},w_{1}w_{2})/ℂ(w_{1}+w_{2},w_{1}w_{2})} ⊴ ℂ(w_{1} + w_{2}, w_{1}w_{2})[X] spanned by these polynomials is prime (it is of dimension 0 and as char(ℂ) = 0 the isolated primary component is prime (again [7, Theorem 5])) and hence coincides with 𝔓_{(w_{1}^{50},w_{2}^{50},w_{1}w_{2})/ℂ(w_{1}+w_{2},w_{1}w_{2})}. The reduced Gröbner basis w. r. t. the graded reverse lexicographic term order where X_{1} ≺ X_{2} ≺ X_{3} is

{X_{1}^{2} − (w_{1}^{50} + w_{2}^{50})X_{1} + (w_{1}w_{2})^{50}, X_{3} − w_{1}w_{2}, X_{2} + X_{1} − w_{1}^{50} − w_{2}^{50}} .

Taking the coefficients as generators we obtain ℂ(w_{1}, w_{2})^{D_{100}} = ℂ(w_{1}^{50} + w_{2}^{50}, w_{1}w_{2}) as one would expect.

5. Implicitization without tag variables

If k is an infinite field and w_{1}, …, w_{l} are algebraically independent over k we may think of x_{1}, …, x_{m} ∈ k(w) as rational functions in the indeterminates w parametrizing the set

{(x_{1}(α), …, x_{m}(α)) : α ∈ k^{l} and x_{1}, …, x_{m} defined at α} .

Using the above notation the implicitization problem in computer aided geometric design can be stated as: “Determine a generating set of 𝔓_{(x)/k}.”

It ist well-known that this problem can be solved by introducing m additional tag variables (see [3, p. 131], [1]). Using the Chow form as described above we can compute 𝔓_{(x)/k} without making use of tag variables. Moreover, if one is only interested in problems like testing whether certain points are contained in V(𝔓_{(x)/k}) it can be sufficient to solve the following weaker variant of the implicitization problem (see, e. g. [5]): “Determine an ideal ℑ ⊴ k[Z] with V(ℑ) = V(𝔓_{(x)/k}).”

To solve this variant it is sufficient to derive generators for the ideal 𝔄_{(x)/k} from the coefficients of F_{(x)/k}(u; ∑ u_{1j}X_{j}, …, u_{t+1j}X_{j}) ∈ k[X][u]. The computation of the associated prime 𝔓_{(x)/k} can be skipped then.

Finally, we want to remark that for efficiency reasons it would be very interesting to know of an alternate solution to the problem of finding the minimal field of definition of an ideal—in spite of the fact that the mentioned proof of Weil shows that Gröbner bases appear quite naturally in this context.

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Received September 8, 1998