Playground

Free exploration - compute Galois groups, build extensions, explore examples

Interactive Playground: Explore Galois Theory

Welcome to the Galois Theory playground! Here you can explore the mathematical structures we've studied throughout this module. Compute arithmetic in finite fields, analyze polynomials and their Galois groups, and visualize the beautiful correspondence between subgroups and intermediate fields. This is where theory meets practice—experiment, discover patterns, and build intuition for the abstract concepts we've covered.

Tool 1: Finite Field Arithmetic Calculator

Perform arithmetic in finite fields 𝔽ₚ and 𝔽_{p^n}. Select a field, choose elements, and compute addition or multiplication. See the complete operation tables and understand how field arithmetic works in both prime fields (like 𝔽₅) and extension fields (like 𝔽₄ = 𝔽₂[x]/(x² + x + 1)). Every calculation follows the field axioms, and every nonzero element has a multiplicative inverse!

Select a Finite Field:

Field: 𝔽₂

Type: Prime field (p = 2)

Size: 2 = 2

Elements: 0, 1

Compute in 𝔽₂:

Operation:

First Element:

Second Element:

Result:

0 + 0 = 0

Addition Table

+	0	1
0	0	1
1	1	0

Multiplication Table

×	0	1
0	0	0
1	0	1

Understanding Finite Field Arithmetic:

Prime fields: In 𝔽ₚ, we work modulo p. For example, in 𝔽₅: 3 + 4 = 7 ≡ 2 (mod 5), and 3 × 4 = 12 ≡ 2 (mod 5).

Extension fields: In 𝔽_{p^n}, elements are polynomials in α, reduced modulo an irreducible polynomial. For example, in 𝔽₄ = 𝔽₂[x]/(x² + x + 1): α × α = α² ≡ α + 1 (since α² + α + 1 = 0).

Notice that every nonzero element has a multiplicative inverse! This is what makes these structures fields.

Tool 2: Polynomial & Galois Group Explorer

Explore famous polynomials and their Galois groups. See how polynomials factor over different fields (ℚ, ℝ, ℂ), discover their roots, and learn about their splitting fields. Compare solvable polynomials (like x⁵ - 1 with abelian Galois group) with unsolvable ones (like x⁵ - x - 1 with Galois group S₅). This tool brings together everything from irreducibility to solvability!

Select a Polynomial to Explore:

Square Root of 2

⋯

Basic Properties:

Degree: 2

Base Field:

…

Irreducible: Yes

Galois Group:

Group: ℤ/2ℤ ≅ S₂

Order: 2

Splitting Field:

…

Degree: [ℚ(√2) : ℚ] = 2

Roots:

…

Factorizations over Different Fields:

Over ℝ (reals):

…

Over ℂ (complex numbers):

…

Key Properties:

Irreducible over ℚ

Splits over ℝ

Minimal polynomial of √2

Abelian Galois group

Explanation:

The polynomial x² - 2 is the minimal polynomial of √2. It has Galois group ℤ/2ℤ (just swapping the two roots). This is the simplest nontrivial field extension.

Understanding Polynomial Properties:

Irreducibility: A polynomial is irreducible over a field F if it cannot be factored into smaller polynomials with coefficients in F. For example, x² - 2 is irreducible over ℚ but factors over ℝ.

Splitting Field: The smallest field extension where the polynomial completely factors into linear terms (degree-1 polynomials).

Galois Group: The group of automorphisms of the splitting field that fix the base field. Its order equals the degree of the splitting field extension (when the extension is Galois/normal).

Tool 3: Galois Correspondence Visualizer

Witness the Fundamental Theorem of Galois Theory in action! Explore the one-to-one correspondence between subgroups of the Galois group and intermediate fields. See how larger subgroups correspond to smaller fields (the lattice is upside-down!), and discover which subgroups are normal (corresponding to Galois intermediate extensions). This is the crowning achievement of the theory—a perfect bridge between algebra and field theory.

Select a Galois Extension:

ℚ(√2) / ℚ

Polynomial:

…

Field Extension:

Base Field:

…

Splitting Field:

…

Degree: [ℚ(√2) : ℚ] = 2

Galois Group:

Group:

…

Order: 2

Abelian: Yes

Cyclic: Yes

Overview:

The simplest nontrivial Galois extension. The Galois group has 2 elements: the identity and the automorphism that swaps √2 and -√2. There are 2 subgroups and 2 intermediate fields.

The Galois Correspondence

Fundamental Theorem: There is a one-to-one correspondence between subgroups of the Galois group and intermediate fields. Larger subgroups correspond to smaller fields!

Subgroup H ≤ Gal(E/F)	Order \|H\|	Index [G:H]	Fixed Field E^H	Degree [E^H:F]	Normal?
… identity only	1	2	…	2	✓
… {e, σ} where σ: √2 ↦ -√2	2	1	…	1	✓

Key Observations:

• Order |H| × Degree [E^H : F] = 2 (the total extension degree)

• Index [G : H] = Degree [E^H : F] (the degree of the fixed field)

• H is normal ⟺ E^H / F is Galois (normal subgroups ↔ Galois intermediate extensions)

Lattice Structure:

Linear lattice: {e} ⊆ ℤ/2ℤ corresponds to ℚ ⊆ ℚ(√2).

Note: The subgroup lattice is upside down from the field lattice! The largest subgroup (the whole Galois group) corresponds to the smallest field (the base field).

The Power of the Galois Correspondence:

The Fundamental Theorem of Galois Theory gives us a dictionary between field theory and group theory:

Field Theory ⟷ Group Theory

Intermediate field E^H ⟷ Subgroup H

[E^H : F] = [G : H] ⟷ Index equals degree

E^H / F is Galois ⟷ H is normal in G

E^H₁ ⊆ E^H₂ ⟷ H₁ ⊇ H₂ (reversed!)

This correspondence:

• Is one-to-one (bijection)

• Is order-reversing

• Preserves normality/Galoisness

• Makes hard field questions easy group questions!

Experiment and Discover

These interactive tools let you explore Galois Theory at your own pace:

Field Calculator

Compute in 𝔽₂, 𝔽₃, 𝔽₅, 𝔽₄, 𝔽₈, 𝔽₉ and see how finite field arithmetic works. Notice that the multiplication table always forms a group (excluding 0).

Polynomial Explorer

Analyze 8 famous polynomials from x² - 2 to the unsolvable quintic x⁵ - x - 1. Compare their Galois groups and understand why some are solvable and others aren't.

Galois Correspondence

See the Fundamental Theorem in action with 4 complete examples. Trace subgroups to their fixed fields and verify the degree formulas.

Suggested Explorations:

1.In the Field Calculator, try computing α² in 𝔽₄. Notice that α² = α + 1 (because α² + α + 1 = 0 in 𝔽₄).

2.In the Polynomial Explorer, compare x³ - 2 (Galois group S₃) with x³ - 3x - 1 (Galois group A₃). Why is one S₃ and the other only A₃?

3.In the Galois Correspondence, look at ℚ(∛2, ω) / ℚ. Notice that the three non-normal subgroups H₁, H₂, H₃ correspond to non-Galois extensions ℚ(∛2), ℚ(ω∛2), ℚ(ω²∛2).

4.Check the formulas: |H| × [E^H : F] should always equal the total extension degree. Verify this in each row of the correspondence tables!

Building Mathematical Intuition

Mathematics is best learned through exploration and experimentation. These tools let you test ideas, verify formulas, and discover patterns on your own. Don't just read about Galois Theory—play with it! Try different fields, different polynomials, different extensions. See what's similar and what's different. Build intuition for the abstract structures. That's how mathematicians really learn.

Next: Ready to test your understanding? Take the comprehensive Galois Theory quiz!