If $F$ is a field of characteristic $p$ prime, how can one create a field $K$ such that $K$ is created from $F$ (either by modding out or by taking a product which includes $F$ or by some other method which involves $F$) such that $K$ has a different characteristic $j \ne p$? Also, does this question depend on $p$? I would also like to consider creations which are field extensions of $F$. Also, how many of each kind of extension are possible?



Field constructions
It should be noted that the characteristic of a field is either prime or zero. If it is zero, then it contains the rational numbers. These are two statements you can probably prove even if algebra isn't your cup of tea. 


You can study valuation rings of mixed characteristic. A classic example is $\mathbb{Z}\_p$ the p-adic integers. This is a ring of characteristic 0 and its fraction field $\mathbb{Q}\_p$ is a field of characteristic 0. However, $\mathbb{Z}\_p$ has a (unique) maximal ideal generated by $(p)\mathbb{Z}\_p$ such that $$\mathbb{F}\_p \cong \mathbb{Z}\_p/(p)\mathbb{Z}\_p$$ which is a finite field of characteristic $p$. 


There is the famous Ax-Kochen theorem, which is the following 


$$\Pi\_\mathcal{F} \mathbb{Q}\\_p \cong \Pi\_\mathcal{F} \mathbb{F}\_p((t))$$


where $\mathcal{F}$ is a non-principal ultra-filter on $\mathbb{N}$. This result depends on the **continuum hypothesis.** (J. Ax and S. Kochen, Diophantine problems over local fields I, American Journal of Mathematics,87 (1965), 605–630.)


However, there is also an isomorphism 


$$\Pi\_\mathcal{F} \mathbb{Z}\\_p \cong \Pi\_\mathcal{F} \mathbb{F}\_p[[t]]$$


where $\mathcal{F}$ is a non-principal ultrafilter on $\mathbb{N}$, which does not depend on the Continuum Hypothesis, in
"Use of Ultrapoducts in Commutative Algebra" by Hans Schoutens, found here
<http://www.springer.com/mathematics/algebra/book/978-3-642-13367-1>


*Witt Vectors* allow you to move from pure characteristic (either zero or positive) to mixed characteristic.


Another way of studying different characteristics is through Algebraic Geometry, by thinking of fields of different characteristics as fibers living over primes in $\mathbb{Z}$. 


**Edit:** It occurred to me that an example here might be useful in order to illustrate this point. Given a curve $C$ over a field of characteristic p>0, we know (Winter's theorem) that there exists a discrete valuation ring (for example, $\mathbb{Z}\_p$ of mixed characteristic), call it B, and a family of curves over B. Over the generic fiber, we have a GAGA principle, which means that things we can define over $\mathbb{C}$ (e.g., fundamental group), we can define over a field of $char(k)=0$. Now using Winter's theorem we can 'transfer' our definitions from analytic geometry to algebraic geometry over a field of positive characteristic, which gives us new tools when studying number theory. 


Finally, there is even an "physical" way to interpret moving between fields. I highly recommend this article of A. Connes on the topic: 
"Characteristic one, entropy and the absolute point," Connes & Consani
<http://arxiv.org/abs/0911.3537>


