what to say about spanning set of finite field extension

Question

Let $F$ is a field.let extension $K$= $F(a_1,a_2,...,a_n)$ where each $a_i$ is algebric over $F$.then $K$ have finite extension over $F$.my problem is can we say something about spanning set of $K$ over $F$

Answer 1

Suppose $F=\mathbb{Q}$ and $K=\mathbb{Q}[\sqrt2,\sqrt3]$. Then the spanning set of $K/F$ is $\{1,\sqrt2,\sqrt3,\sqrt6\}$. The observation here is that elements of $K$ are polynomials in $\sqrt2$ and $\sqrt3$.

Now if $K=F(a_1,...,a_n)$, then elements of $K$ are polynomials in $a_1,...,a_n$ and since each of $a_i$ is algebraic over $F$, so we know that the set $\{1,a_i,(a_i)^2,...(a_i)^{m_i-1}\}$ is a basis for $F(a_i)/F$ where $m_i$ is the degree of $a_i$ over $F$. So can you form a spanning set for $K/F$ now ?

Answer 2

The size of the spanning set has an obvious bound in terms of degrees of the extensions on adjoining $a_i$'s. Can you see it?