MHB 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 $$\mathbb{F}$$ 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 $$\mathbb{F}$$ has characteristic $p$ then, $F_0=\{m\cdot 1:m\in\mathbb{N}\}$ is a subfield of $\mathbb{F}$ somorphic to $\mathbb{Z}/(p)$.