Math 2.0

Commutative algebra Chain conditions Theorem

Hilbert's basis theorem

Builds on Noetherian ringPolynomial ringPolynomials in several variables

Statement

Let RR be a Noetherian commutative ring. Then the polynomial ring R[x]R[x] is Noetherian.

Remarks

By induction, R[x1,…,xn]R[x_1, \dots, x_n] is Noetherian for every nn. In particular, every ideal of K[x1,…,xn]K[x_1, \dots, x_n] for a field KK is finitely generated. The proof studies the ideals of the leading coefficients of the polynomials of each degree in an ideal of R[x]R[x]. In algebraic geometry, the theorem says that every set of polynomial equations has the same solutions as a finite subset of it.

History

David Hilbert proved the theorem in 1890, for polynomials in several variables over a field, in an article on invariant theory. His proof was not constructive. It did not give a method to find the finite set of generators.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Ascending chain condition, definition, no evidence levelAscending chain condi…definitionGroup, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionIdeal, definition, no evidence levelIdealdefinitionPolynomials in several variables, definition, no evidence levelPolynomials in severa…definitionIdeal generated by a subset, definition, no evidence levelIdeal generated by a …definitionNoetherian ring, definition, no evidence levelNoetherian ringdefinitionHilbert's basis theorem, statement, no evidence levelHilbert's basis theor…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Ascending chain condition definition
  2. Group concept
  3. Ring definition
  4. Commutative ring definition
  5. Polynomial ring definition
  6. Ideal definition
  7. Polynomials in several variables definition
  8. Ideal generated by a subset definition
  9. Noetherian ring definition
  10. Hilbert's basis theorem statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_basis_theorem, Statement
  2. Web https://en.wikipedia.org/wiki/Hilbert%27s_basis_theorem, lead section
About this page
Tag0YKA9J3XEE
URIm2://0YKA9J3XEE
Kindstatement, theorem
WritingW1 stated
MSC13E05
StatusThis copy of the site has no build data. Run m2 status 0YKA9J3XEE.
UsesNoetherian ring, Polynomial ring, Polynomials in several variables
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