Integral Extensions and Quadratic Extensions
Handbook guide to integral extensions and quadratic extensions with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Integral Extensions
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Quadratic Extensions of the Rationals
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definitions and Comments In this chapter, unless otherwise specified, all rings
In this chapter, unless otherwise specified, all rings are assumed commutative. Let A be a subring of the ring R, and let x ∈R. Call x is integral over A if x is a root of a monic polynomial f with coefficients in A. The equation f(X) = 0 is called an equation of integral dependence for x over A.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
Let A be a subring of R, with x ∈R. The following conditions are equivalent: (i) x is integral over A; (ii) The A-module A[x] is finitely generated; (iii) x belongs to a subring B of R such that A ⊆B and B is a finitely generated A-module.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Lemma
Let A be a subring of R, with x1, . . . , xn ∈R. If x1 is integral over A, x2 is integral over A[x1],. . . , and xn is integral over A[x1, . . . , xn−1], then A[x1, . . . , xn] is a finitely generated A-module.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Transitivity of Integral Extensions Let A, B and C be subrings of R. If C is
Transitivity of Integral Extensions Let A, B and C be subrings of R. If C is integral over B, that is, each element of C is integral over B, and B is integral over A, then C is integral over A.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Definition
If A is a subring of R, the integral closure of A in R is the set Ac of elements of R that are integral over A. Note that A ⊆Ac because each a ∈A is a root of X −a. Call A is integrally closed in R if Ac = A. If we simply say that A is integrally closed without reference to R, we assume that A is an integral domain with quotient field K, and A is integrally closed in K.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
The integral closure Ac of A in R is integrally closed in R.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Proposition
If A is a UFD, then A is integrally closed.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Definition
A number field is a subfield L of the complex numbers C such that L is a finite extension of the rationals Q. Thus the elements of L are algebraic numbers. The integral closure of Z in L is called the ring of algebraic integers (or simply integers) of L. In the next section, we will find the algebraic integers explicitly when L is a quadratic extension.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Lemma
If a and b are rational numbers, then a + b √ d is an algebraic integer if and only if 2a and a2 −db2 belong to Z. In this case, 2b is also in Z.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Corollary
The set B of algebraic integers of Q( √ d), d square-free, can be described as follows. (i) If d ̸≡1 mod 4, then B consists of all a + b √ d, a, b ∈Z; (ii) If d ≡1 mod 4, then B consists of all u 2 + v 2 √ d, u, v ∈Z, where u and v have the same parity (both even or both odd). [Note that since d is square-free, it is not divisible by 4, so the condition in (i) is d ≡2 or 3 mod 4.]
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Theorem
Let B be the algebraic integers of Q( √ d), d square-free. (i) If d ̸≡1 mod 4, then 1 and √ d form an integral basis of B; (ii) If d ≡1 mod 4, then 1 and 1 2(1 + √ d) form an integral basis.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Quick-reference relationships
Problem-solving workflow
Fix the coefficient ring and variance
State whether modules are left/right modules and whether a functor is covariant or contravariant.
Write the maps, not just the objects
Kernels, images, exactness and universal properties depend on the actual homomorphisms.
Use the appropriate universal property
Direct sums, products, tensor products, projectives, injectives and limits are best handled by their mapping property.
Check exactness at each position
Verify image equals kernel rather than relying on the appearance of a diagram.
Choose a resolution only when needed
Derived constructions should be tied to projective or injective resolutions and independence from the chosen resolution.
Test naturality and compatibility
For induced maps, ensure compositions and commutative squares behave as required.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test ideal, polynomial, basis, field, module, injective, prime, Tor. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Forgetting the coefficient ring when comparing modules.
- Assuming tensor product preserves every exact sequence.
- Confusing direct sum with direct product for infinite families.
- Reading exactness from a diagram without checking image equals kernel.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 7.1 | Integral Extensions | 131–133 |
| 7.2 | Quadratic Extensions of the Rationals | 134–134 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.