# Math Help - Two difficult proofs (Fields)

1. ## Two difficult proofs (Fields)

1.-Prove that each subfield of the field of complex numbers contains every
rational number.

2.-Prove that each field of characteristic zero contains a copy of the rational
number field.

Thanks

2. 1: Let $F$be a field. $0,1\in F\Rightarrow N\subseteq F\Rightarrow -N\subseteq F\Rightarrow Z\subseteq F\Rightarrow \frac{1}{Z^*}\subseteq F\Rightarrow \frac{p}{q}\in F\Rightarrow Q\subseteq F$
2:let $F$be a field, $e$be its identity. Let $F'=\{x|x\in F,x=pe(qe)^{-1},p,q\in Z,q!=0\},\phi:F'\rightarrow Q$to be $\phi(ne)=n, \phi(pe(qe)^{-1})=\frac{p}{q}(q!=0,n,p,q\in Z)$
$ne$ are distinct from each other since if $ne=me$ then $(n-m)e=0\Rightarrow (n-m)a=0\forall a\in F$, which will contradict to $char(F)=0$
it is easy to show that $F'$is a subfield of $F$ and $\phi$is a isomorphism.