Algebraic extension of field with characteristic 0

Question

Is it true that any algebraic extension of a field of characteristic 0 also has characteristic 0? Since the field would be infinite, I assume the amount of algebraic integers in it would also be?

Answer 1

Every field extension preserves the characteristic.

For example, you can't have a homomorphism $f:F\to K$ (injective or not) if $F$ has characteristic $0$ and $K$ has characteristic $p$. Namely, we would have $$f(p) = f(\underbrace{1_F+1_F+\cdots + 1_F}_{p\text{ times}}) = \underbrace{1_K+1_K+\cdots +1_K}_{p\text{ times}} = 0_K $$ but then $$1_K = f(1_F) = f(pp^{-1})=f(p)f(p^{-1}) = 0_K f(p^{-1}) = 0_K $$ which is impossible.