# How to handle points in extended finite field

Following the response to my previous question, I would like to know if you could give me some information or give me a link on how to perform arithmetic operations once I changed a point from the field $\mathbb{F}_p$ to $\mathbb{F}_{p^2}$ ?
If I have an elliptic curve $Y^2 = X^3 + 1$ over $\mathbb{F}_p$ and I want to make a distortion map $\phi(x,y) \rightarrow (\beta x,y)$ to have a point of the form $\{a+bi : a,b \in \mathbb{F_p}\}$ in $\mathbb{F}_{p^2}$.
How can I compute the addition of two points in $\mathbb{F}_{p^2}$ for example since the field arithmetic is not the same ?
As you note, the elements of $\mathbf F_{p^2}$ can be represented as $a+bi$ with $a,b \in \mathbf F_p$ and $i^2 = -1$; however note that this only works if $p\equiv 3 \pmod 4$.
Then you can add and multiply two elements of $\mathbf F_{p^2}$ in basically the same way you add and multiply complex numbers. Namely, $(a+bi)+(c+di) = (a+c)+(b+d)i$ and $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$.
And the formula for point addition is the same, but you need to keep in mind that all your "numbers" are elements of $\mathbf F_{p^2}$ and must be handled as above.