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 and that is isomorphic to either or for some prime p. (Note: 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 be the multiplicative identity in a field of characteristic zero. The subfield generated by is isomorphic to via the map .

__Case 2__: Characteristic , prime

Let be the multiplicative identity in a field of characteristic . The subfield generated by is isomorphic to via the map .

The detailed way to do it:

I'll drop the bar notation from .

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

Now we construct explicitly. Let us start with the element 1. Since (sub)fields are closed under addition, you have , so the natural numbers are contained in .

Next, by definition of subfield. We can write and , which recovers the rest of the integers.

For non-zero elements, inverses exists and are themselves elements of the subfield. In particular, since for any integers (b non-zero) and every rational number is of the form , we deduce that . As such, is a subfield of . It follows that to avoid contradicting the minimality of (remember is the smallest subfield of ).

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 and , then identify with its image 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 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 generated by a set is the intersection of all subfields of containing . In this case, . That paragraph further states that this is the smallest subfield containing . If you don't want to take their word for it, just note that if contains , then for some indexing of all subfields of containing . Since every subfield must contain the multiplicative identity, is the smallest subfield of .