I am studying the structure of complete discrete valued fields. Let $K$ be one such field and denote by $\overline{K}$ the residue field of $K.$ I came across the fact that the following are the only possibilities in terms of the characteristic of $K$ and $\overline{K}$: the equal characteristic case, where both $K$ and $\overline{K}$ have characteristic either $0$ or $p$ ($p$ is prime), and the mixed characteristic case where $K$ has characteristic $0$ and $\overline{K}$ has characteristic $p$.
My question: for the mixed characteristic case, why can’t we have the characteristics the other way around, with $\,\text{char}(K)=p$ and $\,\text{char}(\overline{K})=0$?
Best Answer
This is easier than it might seem since it is more an exercise in definitions. But I think that going through the definitions can't hurt.
The Residue field is defined as the quotien of the valuation ring, i.e. the subring of K whose elements have valuation bigger or equal to 0, quotiented out by the (unique) maximal ideal of this ring, which is made up by the elements whose valuation is bigger or equal to 1.
Now, Since it is a subring of K, the valuation ring has the same charateristic, hence in this case it is of charasteristic p, which is to say that $1+1+...+1$, where the sum has $p$ summands, is $0$. But now you see that this must hold for any quotient of the ring.
Note: We do not use the completeness hypothesis since this holds anyway.