 Solution

Now suppose $\mathrm{char}K=p>0$. The Teichmuller lifts constructed in (b) are additive as well as multiplicative: if $a^q=a$ and $b^q=b$, then the characteristic-$p$ Frobenius identity gives $(a+b{)}^q=a^q+b^q=a+b$. Uniqueness of the lift with prescribed residue yields $[\bar{a}+\bar{b}]=[\bar{a}]+[\bar{b}]$. Their set is therefore a finite coefficient field in positive-characteristic local fields, and reduction identifies it with $k$.

Choose a uniformizer $\pi$. For $x\in {\mathcal{O}}_K$, let $a_0$ be the lift of its residue and write $x=a_0+\pi x_1$ with $x_1\in {\mathcal{O}}_K$. Repeat with $x_1$, then $x_2$, and so on. This givesThe remainder tends to zero, and completeness gives $x={\sum }_{j\ge 0}{a}_j{\pi }^j$. The expansion is unique: in a nonzero difference of two series the first different coefficient is a nonzero element of the coefficient field, hence a unit, and so gives exactly the valuation of that difference.

Consequently substitution $t\mapsto \pi$ defines a bijection $k[[t]]\to {\mathcal{O}}_K$. It is a ring homomorphism because the coefficient lift is a field homomorphism, polynomial operations agree before passage to limits, and multiplication is continuous. Every element of $K$ becomes integral after multiplication by a sufficiently large power of $\pi$. Allowing finitely many negative powers therefore extends substitution to the field isomorphismIt preserves the normalized discrete valuation, since the first nonzero coefficient determines the valuation on each side. The results used are the structural facts stated in (a), its proved Hensel lemma, the proved multiplicative lifting in (b), and the characteristic-$p$ identity $(a+b{)}^q=a^q+b^q$. No classification theorem for local fields is needed.