Is $K = \bigcup\limits_{i:\text{even}} \mathbb{F_{2^i}}$ a field?

Question

1. Is $K = \bigcup\limits_{i:\text{even}} \mathbb{F_{2^i}}$ a field?
2. If so what is char(K)?
3. Is $K \cup \mathbb{F_{2^8}}$ a field?

Answer 1

Hint:

For any prime $p$ and any positive integers $i,j$, $\;\mathbf F_{p^i}\subseteq \mathbf F_{p^j}\;$ if and only if $i\mid j$.

So you essentially have to show the sum and product of two elements of $K$ are well-defined, even if they live in different $\mathbf F_{p^i}$ and $\mathbf F_{p^j}.$ For this, observe that both fields are contained in $\mathbf F_{p^{\operatorname{lcm}(i,j)}}$.

As to the characteristic question, note that all these fields contain $\mathbf F_p$.