In my algebra book they define for field extensions $L/K_1/F$ and $L/K_2/F$ the field $K_1K_2 = K_1(K_2)$.
The formal definition I have for this is $ K_1(K_2) = \bigcap_{K_2 < E < L} E \quad \text{where } E/K_1. $ So the smallest field extension of $K_1$ containing $K_2$.
For extensions such as $F(\alpha)$ though sometimes it is written, $ F(\alpha) = \{ a_0 + a_1\alpha + a_2\alpha^2 + \cdots + a_{n - 1}\alpha^{n - 1} : a_i \in F\} $
Can I extend this definition for $K_1(K_2)$ and write the following? $ K_1(K_2) = \left\{ \sum_{i = 1}^n \sum_{j = 1}^{m_i} a_{ij}b_i^j : a_{ij} \in K_1, b_i \in K_2, n, m_i \in \mathbb{N} \right\}$
EDIT: $K_1, K_2$ are algebraic.