Finding Fraction Field of Z[1/2]

[Metric_Space]

How would I go about finding the fraction field of Z[1/2]?

 

[micromass]

Uuh, wouldn't that just be $\mathbb{Q}$?

In general, if R is an integral domain, and if Q is it's fraction field, then, if

$$R\subseteq S\subseteq Q$$

then the fraction field of S is Q.

 

[Metric_Space]

Yes, that's right -- I guess I don't fully understand how all of these things relate (integral domains, fraction fields, etc)

 

[Metric_Space]

How did you get that so quickly?

 

[micromass]

How did you get that so quickly?

Uuuh, intuition I guess? After a lot of practising, these things come fast...

 

[Metric_Space]

Uuh, wouldn't that just be $\mathbb{Q}$?

In general, if R is an integral domain, and if Q is it's fraction field, then, if

$$R\subseteq S\subseteq Q$$

then the fraction field of S is Q.

Is this a theorem then?

 

[micromass]

It could be, yes...

 

[Metric_Space]

What about Z[1/3]?

 

[micromass]

Doesn't the same theorem apply here?

 

[Metric_Space]

yes, you're right

 

[Metric_Space]

It could be, yes...

do you know the name of this theorem so I could look it up and see the proof?

 

[micromass]

Uuh, I don't know any books that contain the proof. But the proof is a very good exercise. Why not try it for yourself? What do you know about fraction fields? Do you have characterizations for them?

 

[Metric_Space]

Just know the definition :

Fraction field for integral domain = {a/b | a, b are elements of D, b not equal to zero}

 

[micromass]

Can you prove that the fraction field Q of an integral domain A is the smallest field that contains A?

I.e. Assume that K is a field such that

$$A\subseteq K$$

then

$$Q\subseteq K$$

Start by showing this...

Edit: I might have take $\subseteq$ a bit too liberal in the last equation. Formally, there only exists an injective ring morfism $Q\rightarrow K$. But I see that as the same thing as a subset...