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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.00363 |
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Table of Contents:
- In classical mathematics, Gulliksen has introduced the length of Noetherian modules, and Brookfield has determined the length of Noetherian polynomial rings. Brookfield's result can be regarded as a quantitative version of Hilbert's basis theorem. In this paper, based on the inductive definition of Noetherian modules in constructive algebra, we introduce a constructive version of the length called $α$-Noetherian modules, and present a constructive proof of some results by Brookfield. As a consequence, we obtain a new constructive proof of $\dim K[X_0,\ldots,X_{n-1}]<1+n$ and $\dim\mathbb{Z}[X_0,\ldots,X_{n-1}]<2+n$, where $K$ is a discrete field.