Field Notation Question

Question

In a book on abstract algebra I am using for a presentation, I cam across the following theorem:

A real number $\alpha$ is constructible iff there exists a sequence of fields \begin{align*} \mathbb{Q} = F_0 \subset F_1 \subset ... \subset F_k \end{align*} such that $F_i = F_{i-1}[\sqrt{a}]$ for $\alpha_i \in F_i$ and $\alpha \in F_k$. In particular, there exists $k \in \mathbb{Z^+}$ such that $[\mathbb{Q}(\alpha):\mathbb{Q}] = 2^k$

I've never seen the notation $[\mathbb{Q}(\alpha):\mathbb{Q}]$ before and was wondering what it might mean in this context. Any help would be appreciated

Answer 1

Given a field extension $k \subseteq K$ the larger field is always a vector space over the smaller field because:

1. $(K,0,+)$ is an Abelian group
2. Scalar multiplication $k \times K \to K$ makes sense: $a \cdot b = ab$ where the multiplication is that of $K$

We define $[K : k]$ to be the dimension of this vector space: $[K : k] = \dim_k K$.

Think about the following example. Let $k = \mathbf{Q}$ and $K = \mathbf{Q}(i)$ then $[K : k] = 2$ and $\{1,i\}$ is a basis.