# Math Help - Field

1. ## Field

I need the proof:

The set of Rational numbers has no proper sub-field.

2. Q is a prime field of characteristic 0, i.e. Q can be embedded in any field of characteristic 0.
But subfield preserves characteristic.

3. Alternatively, if $F\subseteq\mathbb{Q}$ is a subfield then $1\in F$, so that $1+1,1+1+1,\ldots\in F$ so $\mathbb{N}\subseteq F$. But then $-1,-1-1,\ldots \in F$ so that $\mathbb{Z}\subseteq F$. But for each $0\neq z\in\mathbb{Z}$ we have that $z^{-1} \in F$, so that $pq^{-1} \in F$ for any $p,q\in\mathbb{Z},q\neq 0$. Hence $\mathbb{Q}\subseteq F.$