Rings of Fractions and Fields of Fractions

[Math Amateur]

I am seeking to understand Rings of Fractions and Fields of Fractions - and hence am reading Dummit and Foote Section 7.5

Exercise 3 in Section 7.5 reads as follows:

Let F be a field. Prove the F contains a unique smallest subfield $$F_0$$ and that $$F_0$$ is isomorphic to either $$\mathbb{Q}$$ or $$\mathbb{Z/pZ}$$ for some prime p. (Note: $$F_0$$ is called prime subfield of F.)

I am somewhat overwhelmed with this exercise and need help to get started. Can anyone help with this exercise.

Peter

 

[Fernando Revilla]

Let F be a field. Prove the F contains a unique smallest subfield $$F_0$$ and that $$F_0$$ is isomorphic to either $$\mathbb{Q}$$ or $$\mathbb{Z/pZ}$$ for some prime p. (Note: $$F_0$$ is called prime subfield of F.)

Hints: If \(\displaystyle \mathbb{F}\) has zero characteristic, then $F_0=\{m\cdot 1/n\cdot 1:m\in\mathbb{Z},n\in\mathbb{N^*}\}$ is a subfield of $\mathbb{F}$ isomorphic to $\mathbb{Q}$. If \(\displaystyle \mathbb{F}\) has characteristic $p$ then, $F_0=\{m\cdot 1:m\in\mathbb{N}\}$ is a subfield of $\mathbb{F}$ somorphic to $\mathbb{Z}/(p)$.