Rings of Fractions and Fields of Fractions

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 $\displaystyle F_0 $ and that $\displaystyle F_0 $ is isomorphic to either $\displaystyle \mathbb{Q} $ or $\displaystyle \mathbb{Z/pZ} $ for some prime p. (Note: $\displaystyle F_0 $ is called prime subfield of F.)

Can anyone help with this exercise.

Peter

Re: Rings of Fractions and Fields of Fractions

Are you allowed to assume that the characteristic of a field is either 0 or prime?

Re: Rings of Fractions and Fields of Fractions

D&F exercise 3 section 7.5 makes no assumptions about the Field F

However D&F Chapter 13 page 510 states that "the characteristic of a field is either 0 or a prime p"

Following this, then, you can make the assumption that you mention'

Peter

Re: Rings of Fractions and Fields of Fractions

The slick way to do it, but may not be rigorous:

__Case 1__: Characteristic zero

Let $\displaystyle \overline{1} \in F$ be the multiplicative identity in a field $\displaystyle F$ of characteristic zero. The subfield generated by $\displaystyle \overline{1}$ is isomorphic to $\displaystyle \mathbb{Q}$ via the map $\displaystyle f:\overline{1} \mapsto 1$.

__Case 2__: Characteristic $\displaystyle p$, $\displaystyle p$ prime

Let $\displaystyle \overline{1}\in F$ be the multiplicative identity in a field $\displaystyle F$ of characteristic $\displaystyle p$. The subfield generated by $\displaystyle \overline{1}$ is isomorphic to $\displaystyle \mathbb{Z}/p\mathbb{Z}$ via the map $\displaystyle f:\overline{1} \mapsto 1$.

The detailed way to do it:

I'll drop the bar notation from $\displaystyle \overline{1}$.

The subfield $\displaystyle K$ generated by 1 is defined to be the intersection of all subfields $\displaystyle L\subset F$ containing 1. By the axioms of subfields, every subfield of $\displaystyle F$ contains 1, so $\displaystyle K$ is obviously the smallest subfield of $\displaystyle F$.

Now we construct $\displaystyle K$ explicitly. Let us start with the element 1. Since (sub)fields are closed under addition, you have $\displaystyle n=\underbrace{1+...+1}_{\text{n times}}$, so the natural numbers are contained in $\displaystyle K$.

Next, $\displaystyle -1\in K$ by definition of subfield. We can write $\displaystyle 0=1+(-1)$ and $\displaystyle -n=\underbrace{-1-...-1}_{\text{n times}}$, which recovers the rest of the integers.

For non-zero elements, inverses exists and are themselves elements of the subfield. In particular, since $\displaystyle a,b^{-1}\in K$ for any integers $\displaystyle a,b$ (b non-zero) and every rational number is of the form$\displaystyle \frac{a}{b}\in \mathbb{Q}$, we deduce that $\displaystyle \mathbb{Q}\subset K$. As such, $\displaystyle \mathbb{Q}$ is a subfield of $\displaystyle K$. It follows that $\displaystyle K\cong \mathbb{Q}$ to avoid contradicting the minimality of $\displaystyle K$ (remember $\displaystyle K$ is the smallest subfield of $\displaystyle F$).

NOTE: By abuse of notation, I identified the n-fold sum of 1 with the natural number n. If you want to make things rigorous, write $\displaystyle \overline{1}$ and $\displaystyle \overline{n}$, then identify $\displaystyle \overline{q}\in K$ with its image $\displaystyle q\in \mathbb{Q}$ under the obvious isomorphism.

NOTE 2: The same kind of constructions hold for the characteristic p case - it's even easier actually.

Re: Rings of Fractions and Fields of Fractions

Thanks for your help Gusbob!

Just some points of clarification:

You write:

"The subfield K generated by 1 is defined to be the intersection of all subfields L\subset F containing 1."

This looks more like a proposition or assertion to me rather than a definition. It is easy to show that the intersection of a family of subfields is a subfield but how do you show (rigorously and formally) that "the subfield K generated by 1 is the intersection of all subfields $\displaystyle L \subset F $ containing 1"

You also write "K is the smallest subfield of F". How do you prove this?

Can you help?

Peter

Re: Rings of Fractions and Fields of Fractions

In the same section (7.5) of Dummit and Foote, there is a paragraph before corollary 16 that states that the subfield of $\displaystyle F$ generated by a set $\displaystyle A$ is the intersection of all subfields of $\displaystyle F$ containing $\displaystyle A$. In this case, $\displaystyle A=\{1\}$. That paragraph further states that this is the smallest subfield containing $\displaystyle A$. If you don't want to take their word for it, just note that if $\displaystyle L_k$ contains $\displaystyle A$, then $\displaystyle K=\cap_{i\in I} L_i \subset L_k$ for some indexing $\displaystyle I$ of all subfields of $\displaystyle F$ containing $\displaystyle A$. Since every subfield must contain the multiplicative identity, $\displaystyle K$ is the smallest subfield of $\displaystyle F$.