= Solution
A <local field> is nondiscrete and locally compact. In the non-Archimedean case its <valuation ring> is compact by the scaling argument in part (b). Any <Cauchy sequence> has a tail in a translate of a compact ball, so has a convergent subsequence; the Cauchy property forces the whole sequence to converge. Thus the <field> is complete. Part (b) gives a <discrete valuation>, a <uniformizer> $\pi$, and a finite <residue field> $k=\mathbb F_q$, $q=p^f$.
We construct the <coefficient> <field> rather than presuming it exists. The <polynomial> $T^q-T$ has derivative $-1$ in characteristic $p$. Every residue element therefore has a unique <polynomial root> lift by <Hensel's lemma>. The set $S$ of these $q$ lifts is a <field>: qth powers preserve addition and multiplication, so sums and products remain <polynomial root> lifts; a nonzero lift has inverse $a^{q-2}$. Reduction identifies $S$ with $\mathbb F_q$. This is the <finite coefficient field in positive-characteristic local fields>.
For $x\in\mathcal O_K$, choose $a_0\in S$ with the same residue, write $x=a_0+\pi x_1$, and repeat with $x_1\in\mathcal O_K$. Completeness gives
$$
x=\sum_{j\ge0}a_j\pi^j,\qquad a_j\in S.
$$
The first nonzero <coefficient> determines the <valuation>, so the expansion is unique. Multiplying an arbitrary <field> element by a suitable power of $\pi$ reduces it to this case. Consequently substitution $T\mapsto\pi$ gives a valued-field isomorphism
$$
\boxed{K\cong\mathbb F_{p^f}((T)),\qquad f\ge1.}
$$
Addition and multiplication agree with those of <Laurent series> by convergence, and a nonzero series has a nonzero leading <coefficient>, establishing injectivity as well as surjectivity.
Conversely, $\mathbb F_q((T))$ is complete for its T-adic <absolute value on a field>. Its <valuation ring> $\mathbb F_q[[T]]$ is compact: fixing finitely many <coefficients> gives the finite partitions into balls, and the successive <coefficient> choices realize the <inverse limit> of the finite rings $\mathbb F_q[T]/(T^n)$. Equivalently it is the product of countably many finite discrete <coefficient> sets. Thus it is a non-Archimedean <local field>. Distinct $q$ give nonisomorphic valued <fields> because their <residue fields> have different sizes. Replacing $|T|$ by any number in $(0,1)$ gives an equivalent <absolute value on a field>.
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