# Hilbert's Basis Theorem - Polynomial of Minimal Degree

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Your Name]In summary, the Proof of Hilbert's Basis Theorem states that if R is a commutative noetherian ring, then R[x] is also noetherian. The proof involves choosing a polynomial of minimal degree in I, the ideal in R[x], and using it in an induction process to show that all polynomials in I can be generated by this minimal polynomial. This is possible because the ideal generated by the minimal polynomial includes all other polynomials of the same degree.
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I am reading the Proof of Hilbert's Basis Theorem in Rotman's Advanced Modern Algebra ( See attachment for details of the proof in Rotman).

Hilbert's Basis Theorem is stated as follows: (see attachment)

Theorem 6.42 (Hilbert's Basis Theorem) If R is a commutative noetherian ring, the R[x] is also noetherian.

The proof begins as follows: (see attachment)

Proof: Assume that I is an ideal in R[x] that is not finitely generated; of course [TEX] I \ne 0 [/TEX].

Define [TEX] f_0(x) [/TEX] to be a polynomial in I of minimal degree, and define, inductively [TEX] f_{n+1}(x) [/TEX] to be a polynomial of minimal degree in [TEX] I - (f_0, f_1, f_2, ... ... f_n) [/TEX].

It is clear that [TEX] deg(f_0) \le deg(f_1) \le deg(f_2) \le [/TEX] ... ... ... (1)

Question: Is polynomial of minimal degree simply any polynomial of least or smallest degree in I. If so how can we be sure (1) holds.

If for the stage of choosing [TEX] f_0(x) [/TEX], for example the minimum degree is 3 and there are a number (possibly infinite) of such polynomials in I, how can we be sure that [TEX] (f_0(x)) [/TEX] includes all of these, so that [TEX] I - (f_0(x)) [/TEX] contains no polynomials of degree 3 and so the degree of [TEX] f_1 [/TEX] will be larger and so 1 holds.

To try to answer my own question ... ... I am assuming that the ideal [TEX] (f_0) [/TEX] includes all the polynomials of the same degree as [TEX] f_0(x) [/TEX]. Is that correct?

Can someone clarify the above.

Peter

[This has.also been posted on MHF]

Last edited:

Dear Peter,

Thank you for sharing your thoughts and questions about the Proof of Hilbert's Basis Theorem. I can understand your confusion and I am happy to help clarify the proof for you.

To answer your first question, yes, a polynomial of minimal degree simply means a polynomial of least or smallest degree in I. This is important because the proof relies on the fact that we can always find a polynomial of minimal degree in I, which is then used in the induction process.

Now, to address your second question, you are correct in assuming that the ideal (f_0) includes all the polynomials of the same degree as f_0(x). This is because, by definition, (f_0) is the ideal generated by f_0(x), which means it contains all polynomials that can be obtained by multiplying f_0(x) with other polynomials in R[x]. So, when we choose f_0(x) to be a polynomial of minimal degree in I, we can be sure that (f_0) includes all polynomials of the same degree as f_0(x).

I hope this helps clarify the proof for you. If you have any further questions, please do not hesitate to ask.

## 1. What is Hilbert's Basis Theorem?

Hilbert's Basis Theorem is a fundamental result in abstract algebra that states that every ideal in a polynomial ring over a field has a finite generating set. In other words, every ideal can be generated by a finite number of polynomials.

## 2. What is the significance of this theorem?

The significance of Hilbert's Basis Theorem lies in its applications in various areas of mathematics, such as algebraic geometry, commutative algebra, and algebraic number theory. It is also a crucial tool in proving other important theorems, such as the Nullstellensatz and the Noether normalization lemma.

## 3. Can you provide an example of Hilbert's Basis Theorem in action?

Consider the polynomial ring R = ℚ[x, y, z], where ℚ is the field of rational numbers. Let I be the ideal generated by the polynomials x^2, y^3, and z^4. By Hilbert's Basis Theorem, we can find a finite set of polynomials that generate I. In this case, the set {x^2, y^3, z^4} is a finite generating set for I.

## 4. What is the connection between Hilbert's Basis Theorem and the minimal degree of a polynomial?

Hilbert's Basis Theorem guarantees the existence of a generating set for an ideal, but it does not necessarily provide the most efficient or minimal generating set. The minimal degree of a polynomial in an ideal is the smallest degree among all the polynomials in a generating set for that ideal. Hilbert's Basis Theorem states that there exists a generating set for an ideal with polynomials of minimal degree.

## 5. Are there any limitations to Hilbert's Basis Theorem?

One limitation of Hilbert's Basis Theorem is that it only applies to polynomial rings over fields, and does not hold for polynomial rings over general rings. Another limitation is that it does not provide a method for finding the minimal generating set, and this can be a challenging task in practice.

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