I'm trying to solve this equation and I'm not sure how to proceed. I'm having some difficulties to understand finite field equations with complex numbers.
The task is:
Solve $x^2+4=0$ in $\mathbb{F}_7[\sqrt{-1}]$
My progress so far:
$x^2 = -4$
$x=\sqrt{-1} \cdot \sqrt{4} = {^+_-}2i$
$(x-2i)(x+2i)$
I'm not sure how to continue from here. Do I need to calculate the modulo to proceed with the finite field calculations? Or is ${^+_-}2i$ the final answer?
Thanks a lot in advance.
It might be better to write $i=\sqrt{-1}$, and to keep in mind that in this context, $i$ is not an element of $\mathbb{C}$. What we're doing is we're starting with the field $\mathbb{F}_7$, and we're adding a new element to $i$ to it. And this element $i$ has the property that $i^2=-1$. $\mathbb{F}_7[i]$ is the smallest field which contains both $\mathbb{F}_7$ and $i$.
To solve $x^2+4=0$ in $\mathbb{F}_7[i]$, note that:
$$0=x^2+4=x^2-4i^2=(x-2i)(x+2i).$$
Since $(x-2i)(x+2i)=0$, and we are in a field, it follows that either $x-2i=0$ or $x+2i=0$. So either $x=2i$ or $x=-2i$.