The Galois Correspondence

The fundamental theorem - the beautiful bijection between fields and groups

The Fundamental Theorem of Galois Theory ⭐

This is the crown jewel of Galois Theory! The Fundamental Theorem establishes a stunning correspondence between two seemingly different mathematical objects:

Subgroups H
of the Galois group Gal(E/F)
Intermediate Fields K
where F ⊆ K ⊆ E

Demo 1: The Classic Example -
…

This is the most common example for teaching the Fundamental Theorem. The Galois group is the Klein 4-group, and there are 5 subgroups corresponding to 5 intermediate fields.

Interactive Demo: Click any node in either lattice to see its corresponding element in the other lattice. This is the Fundamental Theorem of Galois Theory in action!

Subgroup Lattice

Subgroups of ℤ/2ℤ × ℤ/2ℤ

Field Lattice

Intermediate Fields

The Fundamental Theorem

The Fundamental Theorem of Galois Theory establishes a beautiful correspondence between:

  • Subgroups H of the Galois group Gal(E/F)
  • Intermediate fields K where F ⊆ K ⊆ E

The correspondence is given by

…
, where E^H is the fixed field of H (elements of E that are fixed by all automorphisms in H).

Key Property:

…

The size of the subgroup equals the degree of the extension from the base field to the fixed field!

Galois Group
⋯
Klein 4-group (non-cyclic group of order 4)
Order:
…
Group Elements:
…
Identity: fixes everything
Order: 1
…
σ(√2) = -√2, σ(√3) = √3
Order: 2
…
τ(√2) = √2, τ(√3) = -√3
Order: 2
…
στ(√2) = -√2, στ(√3) = -√3
Order: 2
Extension:
…
Degree:
…

Understanding the Correspondence

Full Group G: The entire Galois group (all 4 automorphisms) corresponds to the base field ℚ. Why? Because ℚ is fixed by all automorphisms - every element of ℚ stays unchanged.
Subgroup {e, σ}: The automorphisms that fix √3 (identity and σ) correspond to ℚ(√3). Click it to see!
Trivial Subgroup {e}: Only the identity automorphism corresponds to the top field ℚ(√2, √3). Nothing else fixes all elements.
The Pattern: Bigger subgroups → Smaller fixed fields. Smaller subgroups → Bigger fixed fields. The lattices are "upside down" relative to each other!

Demo 2: Simplest Case -
…

The simplest non-trivial Galois extension. Just 2 subgroups and 2 fields. Perfect for understanding the basic correspondence.

Interactive Demo: Click any node in either lattice to see its corresponding element in the other lattice. This is the Fundamental Theorem of Galois Theory in action!

Subgroup Lattice

Subgroups of ℤ/2ℤ

Field Lattice

Intermediate Fields

The Fundamental Theorem

The Fundamental Theorem of Galois Theory establishes a beautiful correspondence between:

  • Subgroups H of the Galois group Gal(E/F)
  • Intermediate fields K where F ⊆ K ⊆ E

The correspondence is given by

…
, where E^H is the fixed field of H (elements of E that are fixed by all automorphisms in H).

Key Property:

…

The size of the subgroup equals the degree of the extension from the base field to the fixed field!

Demo 3: Gaussian Rationals -
…

Adding the imaginary unit i to the rationals. The Galois group is generated by complex conjugation!

Interactive Demo: Click any node in either lattice to see its corresponding element in the other lattice. This is the Fundamental Theorem of Galois Theory in action!

Subgroup Lattice

Subgroups of ℤ/2ℤ

Field Lattice

Intermediate Fields

The Fundamental Theorem

The Fundamental Theorem of Galois Theory establishes a beautiful correspondence between:

  • Subgroups H of the Galois group Gal(E/F)
  • Intermediate fields K where F ⊆ K ⊆ E

The correspondence is given by

…
, where E^H is the fixed field of H (elements of E that are fixed by all automorphisms in H).

Key Property:

…

The size of the subgroup equals the degree of the extension from the base field to the fixed field!

The Formal Statement

Let E/F be a Galois extension with Galois group G = Gal(E/F). Then there is a bijective correspondence:

⋯
Key Properties:
  • The correspondence reverses inclusions:
    …
  • Degree formula:
    …
  • H is normal in G ⟺ E^H/F is a Galois extension
  • If H is normal:
    …

Why This Is So Powerful

The Fundamental Theorem transforms algebraic questions about field extensions into group-theoretic questions about subgroups. This is incredibly powerful because:

1. Counting Made Easy
Instead of finding all intermediate fields (hard!), we can find all subgroups (easier!) and count their sizes.
2. Structure Preserved
The lattice structure is completely preserved. Understanding the group lattice tells us everything about the field lattice!
3. Normality Connection
Normal subgroups correspond to Galois extensions, giving us a way to identify which extensions are "nice".
4. Solvability by Radicals
This correspondence is the key to proving which polynomials can be solved by radicals (the quintic problem!).

🎉 You've Witnessed the Fundamental Theorem!

By clicking through the interactive lattices, you've explored one of the most beautiful results in all of mathematics. This correspondence between algebra and group theory revolutionized our understanding of polynomial equations.

Next: We'll use this powerful tool to understand solvability by radicals and prove why the quintic equation has no general formula!