[proofplan] The proof is a direct application of two previously established results. The hypothesis "absolute ramification index $e = 1$" means $v_A(p) = 1$, so $p$ is a uniformizer of $A$ and $\mathfrak{m}_A = pA$. By the [Mixed Characteristic DVRs with Perfect Residue Field](/theorems/???) characterization, a complete DVR of mixed characteristic with perfect residue field $k$ and uniformizer $p$ is precisely a strict $p$-ring with residue ring $k$. By the uniqueness theorem for [Witt Vectors](/theorems/???), any two strict $p$-rings with the same perfect residue ring are canonically isomorphic via the unique lift of the identity on $k$. Combining: $A \cong W(k)$. The perfectness hypothesis on $k$ is essential in both steps — it enters the characterization theorem and the uniqueness of Witt vector lifts. [/proofplan] [step:Identify $A$ as a strict $p$-ring] Since $A$ is a complete DVR of mixed characteristic $(0, p)$ with absolute ramification index $e = v_A(p) = 1$, the element $p$ is a uniformizer of $A$: $\mathfrak{m}_A = pA$. The residue field $k = A/pA$ is perfect by hypothesis. Mixed characteristic means $\operatorname{char}(A) = 0$ while $\operatorname{char}(k) = p > 0$; the absolute ramification index $e = v_A(p)$ measures how many times $p$ divides itself as a generator of the maximal ideal, so $e = 1$ says $p$ generates $\mathfrak{m}_A$ directly. By the [Mixed Characteristic DVRs with Perfect Residue Field](/theorems/???) theorem, a complete DVR of mixed characteristic has perfect residue field and uniformizer $p$ if and only if it is a strict $p$-ring with $A/pA$ a field. Since $A$ has uniformizer $p$ and $A/pA = k$ is a perfect field, $A$ is a strict $p$-ring. We note that all hypotheses of the characterization theorem are met: $A$ is complete (given), $A$ is a DVR of mixed characteristic (given), $p$ is its uniformizer (from $e = 1$), and $k = A/pA$ is a perfect field (given). The conclusion is that $A$ is a strict $p$-ring. [guided] Recall that a strict $p$-ring is a $p$-torsion-free ring $B$ in which $p$ is not a zero-divisor, the ideal $pB$ is the Jacobson radical, and $B$ is complete and Hausdorff with respect to the $p$-adic topology (i.e., $B \cong \varprojlim B/p^n B$). In our setting, $A$ is a DVR with maximal ideal $\mathfrak{m}_A$. The condition $e = v_A(p) = 1$ means $v_A(p) = 1$, i.e., $p$ generates $\mathfrak{m}_A$. Therefore $\mathfrak{m}_A = pA$, and so the $p$-adic topology coincides with the maximal-ideal topology on $A$. Since $A$ is a complete DVR, it is complete in this topology. The ring $A$ is an integral domain (a DVR is an integral domain), so $p$ is not a zero-divisor. The hypotheses of the [Mixed Characteristic DVRs with Perfect Residue Field](/theorems/???) theorem are thus satisfied: $A$ is a complete DVR, $p$ is its uniformizer, the residue field $k = A/pA$ is perfect, and the characteristic of $k$ is $p$ (since $A$ has mixed characteristic $(0, p)$). The theorem identifies $A$ as a strict $p$-ring with residue ring $k$. [/guided] [/step] [step:Apply uniqueness of Witt vectors to conclude $A \cong W(k)$] The [Witt Vectors](/theorems/???) theorem states: for any perfect ring $R$ of characteristic $p$, there exists a unique (up to isomorphism) strict $p$-ring $W(R)$ with $W(R)/pW(R) \cong R$. We verify the hypotheses for $R = k$: the field $k$ is a perfect ring of characteristic $p$ by assumption. The existence clause gives a strict $p$-ring $W(k)$ with $W(k)/pW(k) \cong k$. Since $A$ is a strict $p$-ring (established in the previous step) with $A/pA \cong k$, both $W(k)$ and $A$ are strict $p$-rings with residue ring $k$. The uniqueness clause of the Witt vectors theorem, made precise by the [Lifting Homomorphisms Between Strict $p$-Rings](/theorems/???) theorem, gives a unique ring isomorphism $\Phi: W(k) \xrightarrow{\sim} A$ reducing to $\operatorname{id}_k$ modulo $p$: \begin{align*} A \cong W(k). \end{align*} The uniqueness of $\Phi$ means there is a canonical isomorphism — no choices involved. This is the content of the "unramified" part: when $e = 1$, the structure of $A$ is completely determined by its residue field $k$ alone, with no additional data (unlike the ramified case $e > 1$, where the extension $A$ depends on the choice of Eisenstein polynomial). [guided] The Witt vector functor $W$ assigns to each perfect ring $R$ of characteristic $p$ a strict $p$-ring $W(R)$ whose underlying set is $R^{\mathbb{N}}$ with multiplication and addition defined by Witt polynomials. The key property is $W(R)/pW(R) \cong R$. The uniqueness statement — used here — says: any two strict $p$-rings $A$ and $B$ with $A/pA \cong B/pB \cong R$ are isomorphic via a unique isomorphism lifting the identity on $R$. This follows from the [Lifting Homomorphisms Between Strict $p$-Rings](/theorems/???): every ring homomorphism $R \to R'$ between perfect characteristic-$p$ rings lifts uniquely to a ring homomorphism $W(R) \to W(R')$ reducing to it modulo $p$. Taking $R = R' = k$ and both maps equal to $\operatorname{id}_k$ gives a unique isomorphism $W(k) \xrightarrow{\sim} A$ over $k$. The hypothesis that $k$ is perfect (not just reduced) is essential here: the Witt vector construction requires perfectness to define the Teichmüller representatives $[a] \in W(k)$ for $a \in k$ (these are the unique multiplicative lifts of elements of $k$), and the lifting theorem requires perfectness to guarantee uniqueness of the ring map $W(k) \to A$. In summary: the proof identifies $A$ canonically as $W(k)$ by checking two conditions — $A$ is a strict $p$-ring (which uses $e = 1$ and completeness) and $A/pA \cong k$ (which is given). The theorem then supplies the unique isomorphism. The conclusion $A \cong W(k)$ is not just an abstract isomorphism: it gives an explicit description of every element of $A$ as an infinite Witt vector $(a_0, a_1, a_2, \ldots)$ with $a_i \in k$, and the ring operations on $A$ correspond to the Witt vector addition and multiplication formulas. This explicit description is the starting point for $p$-adic Hodge theory. For example, if $k = \mathbb{F}_p$, then $W(\mathbb{F}_p) = \mathbb{Z}_p$: the $p$-adic integers are the Witt vectors over $\mathbb{F}_p$. More generally, $W(\mathbb{F}_{p^f})$ is the ring of integers of the unramified extension of $\mathbb{Q}_p$ of degree $f$. The theorem thus classifies all unramified complete DVRs of mixed characteristic over a perfect residue field: they are exactly the rings of Witt vectors $W(k)$. This classification makes precise the sense in which "unramified" means "no additional structure beyond the residue field": every such ring is the unique (up to isomorphism) lift of its residue field to characteristic zero via the Witt vector functor. [/guided] [/step]