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Rings of Fractions and Fields of Fractions

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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 [TEX] F_0 [/TEX] and that [TEX] F_0 [/TEX] is isomorphic to either [TEX] \mathbb{Q} [/TEX] or [TEX] \mathbb{Z/pZ} [/TEX] for some prime p. (Note: [TEX] F_0 [/TEX] 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

Well-known member
MHB Math Helper
Jan 29, 2012
661
Let F be a field. Prove the F contains a unique smallest subfield [TEX] F_0 [/TEX] and that [TEX] F_0 [/TEX] is isomorphic to either [TEX] \mathbb{Q} [/TEX] or [TEX] \mathbb{Z/pZ} [/TEX] for some prime p. (Note: [TEX] F_0 [/TEX] 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 $\mathbb{Z}/(p)$.