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