Hilbert's basis theorem

The Hilbert basis set (after David Hilbert ) is a fundamental theorem in algebraic geometry, it connects various finiteness conditions.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. For more details see Commutative Algebra.

Formulation

The Hilbert basis theorem says in its general form:

• Is a Noetherian ring, then every polynomial with coefficients in noetherian.

Since the algebras of finite type are exactly the quotient rings of polynomial rings, this statement is equivalent to:

• Is a Noetherian ring and a - algebra of finite type, so is noetherian.

The (up to the parlance) in 1888 by Hilbert proved version handles the special case of the body:

• The polynomial ring over a field is Noetherian.

Conclusion

An important application is the following statement: If a subset of a described for a body through an infinite number of polynomial equations, as already satisfy a finite number of them.

Formal: Let be an arbitrary set of polynomials with the set of common zeros (also called Verschwindungsmenge of ):

Then there is a finite number, so that valid

This is the hardest part of the proof of the statement that the Zariski topology is a topology.