How does an integral extension generalize an algebraic field extension?

Over a field, integral and algebraic mean the same thing because monic and arbitrary nonzero polynomials differ only by a unit. Over a general ring the monic condition is essential, capturing the elements that behave like algebraic integers.

Why does Noether normalization matter?

It presents any finitely generated algebra over a field as a finite extension of a polynomial ring, so its dimension equals the number of polynomial variables. This grounds the whole dimension theory of affine varieties in a concrete construction.

Integral Extension

An integral extension is a ring extension in which every element satisfies a monic polynomial over the subring, generalizing algebraic field extensions and controlling how prime ideals relate between the rings.

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Definition

An element of a ring extension is integral over a subring if it is a root of a monic polynomial with coefficients in the subring; the extension is integral when every element is integral, and the integral closure is the set of all such elements.

Scope

This topic covers integral elements and integral dependence, the integral closure of a ring in an extension and normal rings, the lying-over, going-up, and going-down theorems, and Noether normalization, the structural results that found dimension theory.

Core questions

What does it mean for one ring element to be integral over a subring?
What is the integral closure, and when is a ring normal?
How do prime ideals lift and descend along an integral extension?
How does Noether normalization present an algebra as a finite extension of a polynomial ring?

Key theories

Integral closure and normality: The elements integral over a subring form a subring, the integral closure, and a domain equal to its own integral closure in its field of fractions is called integrally closed or normal, a key regularity condition.
Lying-over and going-up theorems: For an integral extension, every prime of the subring is the contraction of a prime of the extension (lying over), and chains of primes lift compatibly (going up), so the prime spectra of the two rings are tightly linked.
Noether normalization: Every finitely generated algebra over a field is a finite, hence integral, module over a polynomial subring in algebraically independent elements, the algebraic heart of dimension theory and of the geometry of affine varieties.

Clinical relevance

Integral extensions are central to algebraic number theory, where the ring of integers of a number field is the integral closure of the integers, and to algebraic geometry, where Noether normalization and the going-up theorem underpin dimension theory and the behavior of finite morphisms between varieties.

History

Integral dependence abstracts the algebraic integers of number theory studied by Dedekind. Emmy Noether's normalization lemma and Krull's work in the 1920s and 1930s made integral extensions the foundation of dimension theory, later geometrically interpreted by Zariski and Grothendieck.

Key figures

Emmy Noether
Wolfgang Krull
David Hilbert
Oscar Zariski

Seminal works

atiyah1969
eisenbud1995
matsumura1989

Frequently asked questions

How does an integral extension generalize an algebraic field extension?: Over a field, integral and algebraic mean the same thing because monic and arbitrary nonzero polynomials differ only by a unit. Over a general ring the monic condition is essential, capturing the elements that behave like algebraic integers.
Why does Noether normalization matter?: It presents any finitely generated algebra over a field as a finite extension of a polynomial ring, so its dimension equals the number of polynomial variables. This grounds the whole dimension theory of affine varieties in a concrete construction.