Critically examine a proof attempting to show the Galois group of Q̄ over Q is uncountable by constructing automorphisms from binary sequences | Step-by-Step Solution

Understanding the Galois Group Analysis Problem

What We're Solving:

We need to critically examine a proof that attempts to show Gal_Q(Q̄) (the Galois group of the algebraic closure of Q over Q) is uncountable by constructing a specific field extension Q(√2, ∛2, ∜2, ...) and trying to define automorphisms using binary sequences.

The Approach:

• 1. Understand what the proposed proof is trying to do
• 2. Examine the mathematical validity of each step
• 3. Identify potential flaws or gaps in the reasoning
• 4. Determine whether the conclusion follows logically

The key insight is that proving uncountability often involves constructing injections from uncountable sets (like binary sequences) into our target set.

Step-by-Step Analysis:

Step 1: Understanding the Field Extension Let K = Q(√2, ∛2, ∜2, ...) = Q(2^(1/2^n) : n ≥ 1). This is indeed a subfield of Q̄, and we can analyze its structure. Each 2^(1/2^n) has degree 2^n over Q when considered individually.

Step 2: Examining the Automorphism Construction The proof likely attempts to:

• Take a binary sequence (a₁, a₂, a₃, ...)
• Define an automorphism σ by σ(2^(1/2^n)) = (-1)^(aₙ) · 2^(1/2^n)
• Claim this gives uncountably many distinct automorphisms

Step 3: Critical Issues to Examine

Issue 1: Field Extension Properties

• Is K/Q actually Galois? We need to check if K is the splitting field of a separable polynomial
• What's the degree [K:Q]? This affects the size of Gal(K/Q)

Issue 2: Well-defined Automorphisms

• Do the proposed maps actually extend to field automorphisms of K?
• We need to verify they preserve all field operations and relations
• Critical problem: The elements 2^(1/2^n) are not independent! We have relations like (2^(1/4))² = 2^(1/2)

Issue 3: The Fundamental Flaw The proposed automorphisms don't respect the multiplicative relations between the generators! If σ(2^(1/2)) = -2^(1/2) and σ(2^(1/4)) = 2^(1/4), then:

• σ((2^(1/4))²) = σ(2^(1/2)) = -2^(1/2)
• But also σ((2^(1/4))²) = (σ(2^(1/4)))² = (2^(1/4))² = 2^(1/2)

This is a contradiction!

Step 4: What Goes Wrong The binary sequences don't correspond to valid field automorphisms because:

• 1. The generators have algebraic dependencies
• 2. Automorphisms must preserve these dependencies
• 3. Most binary sequences would violate these constraints

The Analysis Conclusion:

The proposed proof is flawed. While Gal_Q(Q̄) is indeed uncountable, this particular construction fails because it doesn't account for the algebraic relations between the field generators. The attempted bijection between binary sequences and automorphisms breaks down due to consistency requirements.

A correct proof would need to either:

• Use a different field extension with truly independent generators
• Employ a completely different approach (like using transcendental methods)

Memory Tip:

Remember: "Relations cause restrictions!" When constructing automorphisms of field extensions, you can't choose the images of generators independently if they satisfy algebraic relations. The automorphism must preserve ALL the structure, not just individual elements.

Great question! This type of critical analysis helps you understand not just what makes a proof work, but what subtle issues can make seemingly reasonable arguments fail. Keep questioning the logical steps - that's exactly the right mathematical mindset! 🌟