Results 1 to 6 of 6
Like Tree3Thanks
  • 1 Post By Gusbob
  • 1 Post By Gusbob
  • 1 Post By Gusbob

Thread: Rings of Fractions and Fields of Fractions

  1. #1
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    594
    Thanks
    2

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    Re: Rings of Fractions and Fields of Fractions

    Are you allowed to assume that the characteristic of a field is either 0 or prime?
    Thanks from Bernhard
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    594
    Thanks
    2

    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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    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.
    Last edited by Gusbob; Apr 11th 2013 at 04:07 AM.
    Thanks from Bernhard
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    594
    Thanks
    2

    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
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    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 $\displaystyle F$.
    Thanks from Bernhard
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 8
    Last Post: Mar 29th 2013, 04:52 AM
  2. Replies: 2
    Last Post: Nov 11th 2012, 01:27 PM
  3. Replies: 2
    Last Post: Nov 10th 2012, 10:45 PM
  4. Rings of Fractions - Dummit and Foote Section 7.5
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Nov 10th 2012, 01:08 PM
  5. Replies: 0
    Last Post: Apr 28th 2010, 09:53 AM

Search tags for this page

Search Tags


/mathhelpforum @mathhelpforum