📌 Step 1: Why Do We Need to Extend ℝ?

Imagine you're in the land of real numbers ℝ 🌍, where all numbers are on a number line.

One day, a mathematician asks:

x2+1=0

You try solving for $x$, but no real number works because:

x2=−1

This is impossible in ℝ, because the square of any real number is always non-negative.

🤔 Question:

• What new number must we introduce to solve this equation?

✅ Solution: We invent a new number called $𝚒$ , where:

𝚒²=−1

Now, we define a bigger number system that includes both real numbers and multiples of 𝚒:

ℂ={a+b𝚒∣a,b∈ℝ}

This new world of complex numbers ℂ contains all real numbers plus all imaginary numbers.

💡 Key Idea:

• ℂ is a field extension of ℝ!

📌 What is the Degree of ℂ Over ℝ?

We measure the size of a field extension by its degree.

• Every number in ℂ looks like:

  • $a+bi$, where $a,b∈ℝ$

• There are two basis elements:

  • $1$ (for real numbers)
  • $𝚒$ (for imaginary numbers)

✅ The degree of this extension is:

[ℂ:ℝ]=2

because every complex number can be written as a linear combination of 1 and 𝚒.

💡 Key Idea:

ℂ is a degree 2 extension of ℝ, because it adds exactly one new independent number: 𝚒.

✅ Complex numbers are just another example of a field extension! 🎉

📌 What is the Galois Group of ℂ Over ℝ?

A Galois Group describes how we can swap roots while keeping equations true.

• The equation $x^2+1=0$ has two solutions:

    𝚒 \text{ and } −𝚒

• There's only one symmetry: swapping $𝚒$ with $¬𝚒$.
  • This is called complex conjugation:

a+b𝚒⟷a−b𝚒.

✅ The Galois Group is:

ℤ_2={e,σ}

where:

• $e$ (identity) does nothing.
• $σ$ swaps $𝚒$ with $¬𝚒$.

💡 Key Idea:

The Galois Group of ℂ over ℝ is just two elements, because there's only one way to swap the roots of $x2+1=0$.

🚀 Summary

• ℂ is a field extension of ℝ by adding 𝚒, where $𝚒²=−1$.
• The extension has degree 2 because all numbers are of the form $a+b𝚒$.
• The Galois Group of ℂ/ℝ has 2 elements, swapping 𝚒 with ¬𝚒.
• Field extensions allow us to solve equations that had no solution in the smaller field!