🧩 Puzzle: The Hidden Mirror

🎯 Goal

Understand Galois Theory by thinking of numbers and polynomials as mirrors reflecting different mathematical worlds.

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📖 Setup: The Magical Mirror

You are a sorcerer studying a mysterious magic mirror 🪞 that reflects numbers in unexpected ways.

• Every time you place a number inside the mirror, it transforms according to special rules.
• You soon realize that some numbers remain unchanged, while others move around in cycles.
• The goal is to find out which numbers control the mirror's magic!

The Mirror’s Rule:

• The mirror follows a special equation:

$$ x^3 - 2 = 0 $$

• This means the mirror's reflection is determined by the three solutions to this equation (roots of the polynomial).

Your mission: Find the hidden structure behind the mirror’s reflections! 🔍

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📌 Step 1: Finding the "Reflected" Numbers (Roots of the Polynomial)

The mirror works by reflecting numbers according to an equation.

• To understand its magic, we solve:

  $$[ x^3 - 2 = 0 ]$$

  which means:

  $$[ x = \sqrt[3]{2}, \quad \omega·\sqrt[3]{2}, \quad \omega^2·\sqrt[3]{2} ]$$

  where $( \omega )$ is a complex cube root of unity:

  $$[ \omega = e^{2\pi i / 3} = \frac{-1 + \sqrt{3}i}{2} ]$$

  and it satisfies:

  $$[ \omega^3 = 1 ]$$

  which is also called a complex phasor

  $$[ \omega = 1_{∠𝑘·2𝜋:3} ]$$

  (Just like rotating a shape three times returns it to its original position! 🔄)

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📌 Step 2: Discovering the Symmetries

Looking into the mirror, you notice the reflections are not random—they follow a hidden group structure!

You observe:

• $( \sqrt[3]{2} )$ transforms into $( \omega·\sqrt[3]{2} )$ .
• $( \omega·\sqrt[3]{2} )$ transforms into $( \omega^2·\sqrt[3]{2} )$
• $( \omega^2·\sqrt[3]{2} )$ transforms back into $( \sqrt[3]{2} )$

🌀 This forms a cycle:

$$[ \sqrt[3]{2} \to \omega\sqrt[3]{2} \to \omega^2\sqrt[3]{2} \to \sqrt[3]{2} ]$$

💡 Key Insight: These transformations form a group of symmetries!

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📌 Step 3: Building the Galois Group

Mathematicians discovered that these reflections form a special structure called the Galois Group of the polynomial $( x^3 - 2 )$

👉 The Galois Group is a set of all transformations (permutations) of the roots that preserve algebraic equations.

In our case, the possible transformations (automorphisms) are:

1. Identity $( e )$ (do nothing)
2. $( \sigma )$ Rotating the roots by one step $( (\sqrt[3]{2} \to \omega·\sqrt[3]{2}) )$
3. $( \sigma^2 )$ Rotating by two steps $( (\sqrt[3]{2} \to \omega^2·\sqrt[3]{2}) )$

Since applying $( \sigma )$ three times returns us to the start:

$$[ \sigma^3 = e ]$$

✅ This means the Galois Group of $( x^3 - 2 )$ is cyclic and behaves like $( \mathbb{Z}_3 )$ !

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📌 Step 4: The Deeper Meaning

What Did We Just Discover?

• The solutions to a polynomial have symmetries, just like rotations of a triangle! 🔺
• The Galois Group describes these symmetries mathematically.
• Some polynomials have bigger groups (harder to solve), while some have smaller groups (easier to solve).

🔍 Key Insight: Galois Theory explains why some equations (like quintic equations) cannot be solved using simple formulas!

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🎯 Challenge for You!

Now that you’ve uncovered the hidden symmetries in the mirror, try these:

1. Find the Galois Group of $( x^2 - 2 = 0 )$
2. What is the Galois Group of $( x^4 - 1 = 0 )$ ? (Hint: Think about square roots of $( \pm 1 )$ .)
3. What does it mean if a Galois Group has a simple structure?