Theorem or equation which works on a different field but fails on $\mathbb{R}$

Question

Is there any particular equation which doesn't work on the real plane of numbers but works on other planes?

Answer 1

$1 + 1 = 0$ in $\mathbb{F}_2$

($\mathbb{F}_2$ is the field only consisting of $0$ and $1$ where $1$ is the additive inverse to itself.)

$1+1 \neq 0$ in $\mathbb{R}$

Answer 2

Freshman's Dream, $(x + y)^n = x^n + y^n$, holds in fields of characteristic $n$, when $n$ is prime.

Answer 3

$x^2+1=0$ doesn't admit roots on $\mathbb{R} $, but it does on $\mathbb{C}$