Find counterexamples to the following statements:
In every field $\Bbb F$, if $a\in \Bbb F$, $a+a=0$, then $a=0$;
Counterexample: Consider $\Bbb Z_2$. Let $a = 1$, so $a + a = 2 = 0 \mod 2$. but $a \ne 0$.
In every field $\Bbb F$, $a^2+1 \neq 0 $ for all $a \in \Bbb F $
Counterexample: Consider $\Bbb Z_2$. Let $a = 1$, so $a^2 + 1 = 2 = 0 \mod 2$. Hence $a^2 + 1 = 0 $.
In every field $\Bbb F$, $−1\neq 1$
Counter example: Consider $\Bbb Z_2$. Then $-1 = 1 \mod 2$, and $1 = 1 \mod 2$. Hence $-1 = 1$.
Please improve my answers and thanks in advance!
For both $(1)$ and $(3)$ the only counterexamples are fields of characteristic $2$ (e.g. $\mathbb F_2$, $\mathbb F_{2^m}$, $\mathbb F_2(x)$ with $x$ trascendental, etc.. ).
For $(2)$ you can take every field that contains the $4$-th primitive roots of unity. For example $\mathbb C$, $\mathbb Q(i) $, $\mathbb F_{p^{2s}}$ where $p$ is a prime, etc