I am reading field theory and i can't answer the following:
1.Is $\Bbb R$ algebraic over $\Bbb Q$?
2.If a field is algebraically closed then it has characteristic as $0$.
Obviously $[\Bbb R:\Bbb Q]=\infty $ as $1,\pi,\pi^2,...$ are all linearly independent.But how to prove $1$ in detail?
Regarding the second question, $\Bbb Z_p$ has characteristic as $p$. So I can consider it's algebraic closure .How to prove that algebraic closure of $\Bbb Z_p$ has characteristic as $p$.
Please help.
For 1, which is false, it suffices to state (or prove, but presumably this is proved in the course!) that $\pi$ or $e$ are not algebraic over $\mathbb{Q}$.
For 2, your idea is essentially correct: take the algebraic closure of $F_p$ (or $\mathbb{Z}_p$, which is the same). Any field that extends $F_p$ has characteristic $p$, basically by definition of the characteristic (we add $1$ $p$ times and get $0$, already in $F_p$, so certainly in any larger field).