I am studying Galois Theory and found this exercise:
Describe this subfield of $\mathbb{Z}_2(t): \ \mathbb{Z}_2(t^2)$
The definition of $K(t)$, where $K$ is a field, that I'm using is:
$K(t)=\{\frac{p(t)}{q(t)}: p(t),q(t) \in K[t] \textrm{ and } q\neq 0\}$
So using this definition the only thing I got is that every element of $\mathbb{Z_2}(t)$ it's like this:
$$\sum_{i=1}^{n}\frac{\alpha_it^{2i}+\beta_i}{\gamma_it^{2i}+\delta_i}\textrm{, where }\alpha_i,\beta_i,\gamma_i,\delta_i \in \mathbb{Z_2} \textrm{ and } \gamma_it^{2i}+\delta_i\neq 0 \ $$ But I don't feel it's correct because it's just the definition nothing more.
There's more thing to do ?
Your description of elements of $\Bbb F_2(t)$ is not on the mark. A typical element of this field looks like $$ \frac{a_0+a_1t+a_2t^2+\cdots a_mt^m}{b_0+b_1t+b_2t^2+\cdots b_nt^n}\,, $$ with not all of the $b_j$ being zero.
The typical element of $\Bbb F_2(t^2)$ is same, but with all exponents being even.
I’m sure they wanted you to recognize that the big field is quadratic over the smaller field.