Give an example of a field $\mathbb{K}$ of characteristic $p > 0$ and elements $a,b \in \bar{\mathbb{K}}$, such that $\mathbb{K}(a)$ and $\mathbb{K}(b)$ are isomorphic as vector spaces over $\mathbb{K}$, but not isomorphic as fields.
I was thinking of $\mathbb{F}_2(\sqrt{2})$ and $\mathbb{F}_2(\sqrt{3})$...
Given two finite fields $E_1$ and $E_2$ that are both extensions of a field $\mathbb{K}$, we have that $$E_1\cong E_2 \text{ as fields} \iff \lvert E_1\rvert = \lvert E_2\rvert \iff E_1\cong E_2\text{ as $\mathbb{K}$-vector spaces}$$ so you can't find any examples with $\mathbb{K}$ finite.
Hint: Try using the field $\mathbb{K}=\mathbb{F}_p(T)$. For concreteness, let's use $p=2$. As long as the fields $\mathbb{K}(a)$ and $\mathbb{K}(b)$ have the same dimension as $\mathbb{K}$-vector spaces, i.e., as long as $[\mathbb{K}(a):\mathbb{K}]=[\mathbb{K}(b):\mathbb{K}]$, they will be isomorphic as $\mathbb{K}$-vector spaces. But you can choose $a$ and $b$ to be the roots of some very different irreducible polynomials in $\mathbb{K}[x]$, like say perhaps $x^2-T$ and $x^2+x+1$, and you can prove the resulting fields are not isomorphic.