Are you allowed to assume that the characteristic of a field is either 0 or prime?
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.
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'
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.
Thanks for your help Gusbob!
Just some points of clarification:
"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?
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 .