6.1 Statement and proof of Hilbert's Nullstellensatz

Hilbert's Nullstellensatz is a game-changer in algebraic geometry. It connects the dots between algebraic varieties and ideals in polynomial rings, giving us a powerful tool to study geometric shapes using algebra.

This theorem is like a bridge between algebra and geometry. It lets us switch between looking at common zeros of polynomials and the ideals they generate, opening up new ways to solve tricky math problems.

Hilbert's Nullstellensatz

Statement and Significance

Top images from around the web for Statement and Significance

• 

  Use the degree and leading coefficient to describe end behavior of polynomial functions ... View original

  Is this image relevant?

• 

  Define and Evaluate Polynomials | Intermediate Algebra View original

  Is this image relevant?

• 

  Identifying the Degree and Leading Coefficient of Polynomials | College Algebra View original

  Is this image relevant?

• 

  Use the degree and leading coefficient to describe end behavior of polynomial functions ... View original

  Is this image relevant?

• 

  Define and Evaluate Polynomials | Intermediate Algebra View original

  Is this image relevant?

1 of 3

Top images from around the web for Statement and Significance

• 

  Use the degree and leading coefficient to describe end behavior of polynomial functions ... View original

  Is this image relevant?

• 

  Define and Evaluate Polynomials | Intermediate Algebra View original

  Is this image relevant?

• 

  Identifying the Degree and Leading Coefficient of Polynomials | College Algebra View original

  Is this image relevant?

• 

  Use the degree and leading coefficient to describe end behavior of polynomial functions ... View original

  Is this image relevant?

• 

  Define and Evaluate Polynomials | Intermediate Algebra View original

  Is this image relevant?

1 of 3

• Hilbert's Nullstellensatz, also known as Hilbert's zero-point theorem, establishes a correspondence between algebraic varieties and ideals in polynomial rings
• The theorem states that for any algebraically closed field $k$ and polynomials $f_1, ..., f_n \in k[x_1, ..., x_n]$, the polynomials have a common zero in $k^n$ if and only if the ideal generated by the polynomials is not the entire ring
  • Provides a link between the geometric notion of an algebraic variety (set of common zeros of polynomials) and the algebraic concept of an ideal in a polynomial ring
  • Allows for the study of geometric properties of varieties using algebraic techniques and vice versa
• Generalizes the fundamental theorem of algebra, which states that every non-constant polynomial over the complex numbers has a root
• Named after David Hilbert, who proved a version of the theorem in the late 19th century, although the current formulation is due to later mathematicians

Implications and Applications

• Has significant implications in algebraic geometry, enabling the use of algebraic techniques to study geometric properties of varieties and vice versa
• Establishes a correspondence between maximal ideals in $k[x_1, ..., x_n]$ and points in affine space $k^n$
  • Every maximal ideal is of the form $(x_1 - a_1, ..., x_n - a_n)$ for some point $(a_1, ..., a_n) \in k^n$
• Allows for the application of powerful algebraic tools, such as Gröbner bases, to solve systems of polynomial equations and study their solution sets
• Plays a crucial role in the development of modern algebraic geometry, including the study of schemes and sheaves

Proving Hilbert's Nullstellensatz

Key Algebraic Concepts

• The proof relies on several important algebraic concepts and techniques:
  • Maximal ideals: An ideal $I$ in a ring $R$ is maximal if $I \neq R$ and there is no ideal $J$ such that $I \subsetneq J \subsetneq R$
  • Localization of rings: Constructing a new ring by inverting certain elements of the original ring, used to reduce the problem to the case of a local ring with a unique maximal ideal
  • Rabinowitsch trick: A clever algebraic manipulation that introduces a new variable to transform a system of polynomial equations into a single equation
• The Hilbert basis theorem, which states that every ideal in a polynomial ring over a field is finitely generated, is also used in the proof

Proof Outline

• Show that if an ideal $I$ in $k[x_1, ..., x_n]$ is proper (not the entire ring), then there exists a maximal ideal containing $I$
  • Use Zorn's lemma, which states that if every chain in a partially ordered set has an upper bound, then the set contains at least one maximal element
• Use localization of rings to reduce the problem to the case of a local ring with a unique maximal ideal
• Apply the Rabinowitsch trick to prove the "if" direction of the Nullstellensatz
  • Show that if the polynomials have no common zero, then the ideal they generate must be the entire ring
• Combine these steps with the Hilbert basis theorem to complete the proof of Hilbert's Nullstellensatz

Ideals vs Varieties

Correspondence between Ideals and Varieties

• Hilbert's Nullstellensatz establishes a fundamental correspondence between ideals in polynomial rings and algebraic varieties
• Given polynomials $f_1, ..., f_n \in k[x_1, ..., x_n]$, the algebraic variety $V(f_1, ..., f_n)$ is the set of all points in $k^n$ that are common zeros of the polynomials
• The ideal generated by the polynomials, denoted by $(f_1, ..., f_n)$, consists of all polynomials that can be expressed as linear combinations of the $f_i$ with polynomial coefficients
• The Nullstellensatz states that the ideal $(f_1, ..., f_n)$ is proper if and only if the variety $V(f_1, ..., f_n)$ is non-empty

Studying Ideals and Varieties

• The correspondence allows for the study of geometric properties of varieties using algebraic techniques, such as:
  • Ideal operations (sum, product, intersection, quotient)
  • Gröbner bases, which provide a systematic way to solve systems of polynomial equations and study their solution sets
• Conversely, it enables the study of algebraic properties of ideals using geometric intuition
  • Visualizing varieties as geometric objects in affine space
  • Using geometric arguments to prove algebraic statements about ideals
• The Nullstellensatz implies a one-to-one correspondence between maximal ideals in $k[x_1, ..., x_n]$ and points in affine space $k^n$
  • Every maximal ideal is of the form $(x_1 - a_1, ..., x_n - a_n)$ for some point $(a_1, ..., a_n) \in k^n$
  • Every point in $k^n$ corresponds to a unique maximal ideal