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Math Help - Rings of Fractions and Fields of Fractions

  1. #1
    Super Member Bernhard's Avatar
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    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
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    Re: Rings of Fractions and Fields of Fractions

    Are you allowed to assume that the characteristic of a field is either 0 or prime?
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    Super Member Bernhard's Avatar
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    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
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    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.
    Last edited by Gusbob; April 11th 2013 at 04:07 AM.
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  5. #5
    Super Member Bernhard's Avatar
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    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  L \subset F containing 1"

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

    Can you help?

    Peter
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    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.
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