# Rings of Fractions and Fields of Fractions

• Apr 11th 2013, 03:10 AM
Bernhard
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 $F_0$ 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.)

Can anyone help with this exercise.

Peter
• Apr 11th 2013, 03:52 AM
Gusbob
Re: Rings of Fractions and Fields of Fractions
Are you allowed to assume that the characteristic of a field is either 0 or prime?
• Apr 11th 2013, 03:58 AM
Bernhard
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
• Apr 11th 2013, 05:04 AM
Gusbob
Re: Rings of Fractions and Fields of Fractions
The slick way to do it, but may not be rigorous:

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

Case 2: Characteristic $p$, $p$ prime
Let $\overline{1}\in F$ be the multiplicative identity in a field $F$ of characteristic $p$. The subfield generated by $\overline{1}$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$ via the map $f:\overline{1} \mapsto 1$.

The detailed way to do it:

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

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

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

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

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

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 $\overline{1}$ and $\overline{n}$, then identify $\overline{q}\in K$ with its image $q\in \mathbb{Q}$ under the obvious isomorphism.

NOTE 2: The same kind of constructions hold for the characteristic p case - it's even easier actually.
• Apr 12th 2013, 03:08 AM
Bernhard
Re: Rings of Fractions and Fields of Fractions

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 $L \subset F$ containing 1"

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

Can you help?

Peter
• Apr 12th 2013, 03:30 AM
Gusbob
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 $F$ generated by a set $A$ is the intersection of all subfields of $F$ containing $A$. In this case, $A=\{1\}$. That paragraph further states that this is the smallest subfield containing $A$. If you don't want to take their word for it, just note that if $L_k$ contains $A$, then $K=\cap_{i\in I} L_i \subset L_k$ for some indexing $I$ of all subfields of $F$ containing $A$. Since every subfield must contain the multiplicative identity, $K$ is the smallest subfield of $F$.