I define a Function field in $y$ defined by $y^2 + y + \frac{x^2 + 1}{x}$ over $\mathbb{F}_2$.
Its rational places are
[Place $(\frac{ 1}{x}, \frac{ 1}{x} y)$, Place $(x, x y)$, Place $(x + 1, x y)$, Place $(x + 1, x y + 1)$]
and degree two places are
[Place $(x^2 + x + 1, x y + 1)$, Place $(x^2 + x + 1, x y + x + 1)$]
If I take divisor $G=$ Place $(x^2 + x + 1, x y + x + 1)$, basis of corresponding Riemann Roch space is $[1, (\frac{xy}{(x^2 + x + 1))} + \frac{1}{x^2 + x + 1})]$.
What is the value of $\frac{xy}{(x^2 + x + 1) } + \frac{1}{x^2 + x + 1}+$ Place $(x^2 + x + 1, x y + 1)$ ?
It is an element of residue field which is isomorphic to $\mathbb{F}_{2^2}$. Since $\mathbb{F}_{2^2}$ is isomorphic to $\mathbb{F}^2_{2}$ as a vector space, I want value in $\mathbb{F}^2_{2}$. What is the working rule here in general.