Galois Numbers vs. Abelian Numbers: Algebraic Number Theory

Fundamental Distinction

Galois numbers and Abelian numbers are concepts from algebraic number theory and Galois theory, describing different properties of algebraic number fields and their symmetry groups.

The terms refer not to the numbers themselves, but to the Galois groups of their minimal polynomials or field extensions.

Core Definitions

Galois Numbers / Galois Extension

An algebraic number α is called a Galois number (or lies in a Galois extension) if its minimal polynomial over ℚ splits completely in the field ℚ(α) itself.

Equivalently: The field extension ℚ(α)/ℚ is a Galois extension — it is both normal (all roots of the minimal polynomial are in the field) and separable (no repeated roots in characteristic 0, automatic for ℚ).

Abelian Numbers / Abelian Extension

An algebraic number α is called an Abelian number if it lies in an Abelian extension of ℚ — a Galois extension whose Galois group is Abelian (commutative).

Equivalently: The automorphisms of ℚ(α)/ℚ commute with each other. This is a much stronger condition than just being Galois.

Note: These terms are more commonly used for field extensions rather than individual numbers. When we say "α is a Galois number," we mean "ℚ(α) is a Galois extension of ℚ."

Hierarchy of Field Extensions

Inclusion Relationship

All Algebraic Numbers

⊃

Galois Numbers (Field extensions are Galois)

⊃

Abelian Numbers (Galois group is commutative)

⊃

Cyclotomic Numbers (Special case: roots of unity)

Every Abelian number is a Galois number, but not conversely. Every Galois number is algebraic, but not all algebraic numbers are Galois.

Detailed Comparison

Galois Numbers / Extensions

Key Property: Normal and separable extension

• Definition: ℚ(α)/ℚ is Galois if it's the splitting field of α's minimal polynomial
• Galois Group: Can be any finite group (symmetric, alternating, dihedral, etc.)
• Symmetry: All conjugates of α are in ℚ(α)
• Example Numbers:
  • √2 + √3 (Galois group: V₄, Klein four-group)
  • Roots of x³ - 2 = 0 are NOT Galois over ℚ (splitting field is bigger)
• Fundamental Theorem: Galois correspondence between intermediate fields and subgroups

Abelian Numbers / Extensions

Key Property: Galois extension with commutative Galois group

• Definition: ℚ(α)/ℚ is Abelian if Gal(ℚ(α)/ℚ) is an Abelian group
• Galois Group: Must be commutative (ℤ/nℤ, products thereof, finite Abelian groups)
• Symmetry: All automorphisms commute (order doesn't matter)
• Example Numbers:
  • n-th roots of unity (cyclotomic fields)
  • Numbers in ℚ(√2, √3) (Galois group: ℤ/2ℤ × ℤ/2ℤ)
  • Constructible numbers (nested square roots)
• Kronecker-Weber Theorem: Every finite Abelian extension of ℚ is contained in a cyclotomic field

Mathematical Characterization:

α is a Galois number ⇔ ℚ(α) = splitting field of minpoly_ℚ(α)

α is an Abelian number ⇔ ℚ(α) is Galois over ℚ AND Gal(ℚ(α)/ℚ) is Abelian

Where Gal(ℚ(α)/ℚ) = {σ: ℚ(α) → ℚ(α) automorphism fixing ℚ} with group operation = composition.

Property Comparison Table

Property	Galois Numbers	Abelian Numbers
Group Structure	Galois group can be any finite group (Sₙ, Aₙ, Dₙ, etc.)	Galois group must be commutative (finite Abelian group)
Field Inclusion	ℚ(α) contains all conjugates of α	ℚ(α) contains all conjugates AND automorphisms commute
Example Polynomial	x⁴ - 10x² + 1 (√2+√3) → Group V₄ (Abelian but all Galois)	xⁿ - 1 (roots of unity) → Group (ℤ/nℤ)× (Abelian)
Counterexample	³√2 (x³-2): ℚ(³√2) is NOT Galois over ℚ (missing complex roots)	√2 + ³√2: Galois but group S₃ (non-Abelian) → not Abelian
Solvability by Radicals	Galois solvable ⇔ roots expressible by radicals	Always solvable by radicals (Abelian ⇒ solvable)
Class Field Theory	General Galois theory: subgroup-field correspondence	Explicit classification via Kronecker-Weber: contained in cyclotomic fields

Examples and Non-examples

Example 1: √2 + √3

Minimal polynomial: x⁴ - 10x² + 1

Galois group: V₄ ≅ ℤ/2ℤ × ℤ/2ℤ (Klein four-group, Abelian)

Status: Both Galois AND Abelian (commutative group)

Example 2: ³√2

Minimal polynomial: x³ - 2

Field ℚ(³√2): Contains only real cube root, missing complex roots e^(2πi/3)³√2

Status: NOT Galois (not normal), therefore NOT Abelian

Example 3: Root of x⁵ - x - 1 = 0

Galois group: S₅ (symmetric group on 5 letters)

Status: Galois but NOT Abelian (S₅ is non-commutative)

Note: This polynomial is not solvable by radicals (S₅ not solvable)

Historical Context

Évariste Galois (1811-1832): Developed Galois theory to understand solvability of polynomial equations. Showed that an equation is solvable by radicals if and only if its Galois group is solvable.

Niels Henrik Abel (1802-1829): Proved the insolvability of the quintic. Abelian groups are named in his honor, though he didn't develop group theory per se.

Leopold Kronecker (1823-1891): Kronecker-Weber theorem (proved by Weber and Hilbert): Every finite Abelian extension of ℚ is contained in a cyclotomic field ℚ(ζₙ).

The Fundamental Insight

Abelian extensions are infinitely better understood than general Galois extensions because of the Kronecker-Weber theorem, which provides a complete classification in terms of cyclotomic fields.

For general Galois extensions with non-Abelian groups, no such complete classification exists — this is at the heart of modern research in number theory, including the Langlands program.

Advanced Connections

Class Field Theory

Class field theory is essentially the study of Abelian extensions of number fields. It provides a beautiful correspondence:

Abelian extensions of a number field K ↔ Information about the arithmetic of K (ideal class groups, unit groups, etc.)

This is why Abelian extensions are so special — they admit an "explicit" description in terms of the base field's arithmetic.

Non-Abelian Class Field Theory (Langlands Program)

The Langlands program is a vast generalization attempting to understand non-Abelian Galois extensions through automorphic forms and representations.

While Abelian extensions correspond to characters (1-dimensional representations), non-Abelian extensions correspond to higher-dimensional representations — much more complex and mysterious.

Summary: Key Differences

1. Definition: Galois numbers come from Galois extensions (normal + separable). Abelian numbers come from Galois extensions with commutative Galois groups.
2. Inclusion: All Abelian numbers are Galois, but most Galois numbers are not Abelian.
3. Group Theory: Galois groups can be any finite group; Abelian Galois groups must be commutative.
4. Classification: Abelian extensions of ℚ are completely classified by cyclotomic fields (Kronecker-Weber). No such classification exists for general Galois extensions.
5. Solvability: Polynomials with Abelian Galois groups are always solvable by radicals (indeed, by nested radicals in a particularly simple way). Galois extensions can have non-solvable groups (e.g., S₅ for quintics).

Practical Implication

When you encounter an algebraic number:

• Ask first: "Is ℚ(α) a Galois extension of ℚ?" (Does it contain all conjugates?)
• If yes, then ask: "Is the Galois group Abelian?" (Do all automorphisms commute?)
• If both yes, you have powerful classification theorems (Kronecker-Weber) and explicit description tools at your disposal.

The study of Abelian numbers is essentially "solved" by class field theory, while non-Abelian Galois numbers represent one of the deepest frontiers in modern mathematics.