Constructing an explicit tower with quadratic extensions.

Question

I am so confused as to where to even start with this problem:

a, b, c, and d are real numbers. Let α = a + bi, β = c + di, such that α = β².

Let Q be the set of rational numbers. Let K be a field, such that Q ⊆ K and a, b ∈ K:

K₀ ⊂ K₁ ⊂ K₂ ⊂ ... ⊂ K_r
K = K₀
My job is to find r and construct an explicit tower while considering the following:

All [K_j+1 : K_j ] = 2
c, d ∈ K_r
All K_i contain real elements (thus, we can't extend by i).

I've been trying to work with norms, but I just don't see it right now. I'm hoping to implement the following theorem:
If F and K are both fields, with [F:K] = 2, and characteristic of K not equal to 2, then ∃α ∈ F such that F = K(α) and α² = β ∈ K

A nudge in the right direction would be much appreciated. Thank you for your time!

Answer 1

Consider the sequence of primes $(p_n)_{n \geq 1}$. Then the following is what you want,

$$\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},...) = \bigcup_{n=1}^{\infty} \mathbb{Q}(\sqrt{p_1},\sqrt{p_2},...,\sqrt{p_n})$$