Witt vectors

The ring of Witt vectors is a functor that takes in a ring \( R \) and constructs another ring \( W _ p(R) \). If we take a finite field \( \mathbb{F} _ q \) with \( q = p^n \), it outputs a ring \( W _ p(\mathbb{F} _ q) = \mathcal{O} _ K \) where \( K \) is the unique unramified extension of \( \mathbb{Q} _ p \) of degree \( n \). So in some sense, Witt vectors give a canonical construction of local fields with a specific residue field.

The theory of Witt vectors as developed by Witt [Wit37]. Lang [Lan02] also has a list of exercises leading up to the theory. I have mostly looked at these exercises and tried to figure out the solutions. There is also Rabinoff’s article [Rab14] available, which gives a general treatment of the subject.

Aside from the construction of Witt vectors, we also discuss one application of Witt vectors. In his original paper [Wit37], Witt proves a generalization of Artin–Schreier theory, which classifies cyclic extensions of degree \( p \) over characteristic \( p \) fields. Witt’s generalization works with cyclic degree \( p^n \) extensions, and the construction is given by truncated Witt vectors.

1. Motivation#

Suppose we have finite unramified extension \( K / \mathbb{Q} _ p \), and assume that we want write down an element of \( \mathcal{O} _ K \). Because \( K \) is local with \( p \) as a uniformizer, the natural way to do this is to write an elements \( x \in \mathcal{O} _ K \) as \[ \displaystyle x = \sum _ {n=0}^{\infty} x _ n p^n \] for some representatives \( x _ n \in \mathcal{O} _ K \) of \( \mathcal{O} _ K / p \mathcal{O} _ K = k \). But which representatives should we take? If \( K = \mathbb{Q} _ p \) a natural choice is to take \( \{0, 1, \ldots, p-1\} \) as we do for integers, but doing computations with such a representatives would not be fun, as we have to worry about carrying all the time.

Fortunately, there is a unique homomorphism \[ \displaystyle \tau : k^\times \rightarrow \mathcal{O} _ K^\times; \quad x \mapsto \lim _ {n \rightarrow \infty} x^{\lvert k \rvert^n} \] satisfying the property that \( \tau(x) \) reduces to \( x \) modulo \( p \). (Uniqueness immediately follows from Hensel applied to \( X^{\lvert k \rvert-1} - 1 \).) This map is called the Teichmüller character, and can also be extended to \[ \displaystyle \tau : k \rightarrow \mathcal{O} _ K \] by \( \tau(0) = 0 \). Now let us try to use the image of \( \tau \) as the representatives in the \( p \)-adic expansion. Clearly there exists a unique sequence of \( x _ n \in k \) such that \[ \displaystyle x = \sum _ {n=0}^{\infty} \tau(x _ n) p^n. \]

Now that we have a good way of writing down elements, we want to know how addition and multiplication behaves with this representation. This is going to be a difficult task, because \( \tau \) doesn’t behave well with addition. But we can at least try to figure out the first few lower order terms. If \( x, y \in \mathcal{O} _ K \), \[ \displaystyle x + y = \sum _ {n = 0}^{\infty} (\tau(x _ n) + \tau(y _ n)) p^n = \sum _ {n=0}^{\infty} \tau((x+y) _ n) p^n. \] Working modulo \( p \) immediately gives \[ \displaystyle (x+y) _ 0 = x _ 0 + y _ 0. \] Then modulo \( p^2 \) gives \[ \displaystyle x _ 0^{\lvert k \rvert} + y _ 0^{\lvert k \rvert} + x _ 1 p + y _ 1 p \equiv (x _ 0 + y _ 0)^{\lvert k \rvert} + (x+y) _ 1 p \pmod{p^2} \] and thus \[ \displaystyle (x+y) _ 1 = x _ 1 + y _ 1 - \sum _ {j=1}^{p-1} \frac{1}{p} \binom{\lvert k \rvert}{j \lvert k \rvert / p} x _ 0^{j \lvert k \rvert / p} y _ 0^{(p-j) \lvert k \rvert / p}. \] Here, the Frobenius on \( k \) is an isomorphism, and we can write \( x _ 0^{\lvert k \rvert/p} \) just as \( x _ 0^{1/p} \), where raising to the power of \( 1/p \) simply denotes the inverse Frobenius. This simplifies the formula to \[ \displaystyle (x+y) _ 1 = x _ 1 + y _ 1 - \sum _ {j=1}^{p-1} \frac{1}{p} \binom{p}{j} x _ 0^{j/p} y _ 0^{(p-j)/p}, \] because \( \binom{p^a}{j p^{a-1}} \equiv \binom{p}{j} \bmod{p^2} \). Because we don’t want \( 1/p \) on the exponent, we may apply the Frobenius to both sides and write this as \[ \displaystyle (x+y) _ 1^p = x _ 1^p + y _ 1^p - \sum _ {j=1}^{p-1} \frac{1}{p} \binom{p}{j} x _ 0^{j} y _ 0^{p-j}. \] Note that this is a well-defined polynomial with integer coefficients.

What about multiplication? Again it can be computed that \[ \displaystyle (xy) _ 0 = x _ 0 y _ 0, \quad (xy) _ 1^p = x _ 0^p y _ 1^p + y _ 0^p x _ 1^p, \quad \ldots. \] From these computations, it is natural to expect that \( (x+y) _ n^{p^n} \) and \( (xy) _ n^{p^n} \) can be expressed as integer coefficient polynomials in terms of \( x _ j^{p^j} \) and \( y _ j^{p^j} \) for \( j < n \). This motivates the definition of Witt vectors.

2. Witt vectors#

Consider an infinite sequence of variables \( X _ 0, X _ 1, X _ 2, \ldots \), corresponding to the coefficients \( x _ 0, x _ 1^p, x _ 2^{p^2}, \ldots \) in the previous example. We define the ghost components \[ \displaystyle X^{(n)} = X _ 0^{p^n} + p X _ 1^{p^{n-1}} + p^2 X _ 2^{p^{n-2}} + \cdots + p^n X _ n. \] We now define addition and multiplication so that the ghost components satisfy \[ \displaystyle (X+Y)^{(n)} = X^{(n)} + Y^{(n)}, \quad (XY)^{(n)} = X^{(n)} Y^{(n)}. \] If we could make this work with \( (X+Y) _ n \) and \( (XY) _ n \) being integer coefficient polynomials in \( X _ n \) and \( Y _ n \), then we could make the following definition.

Definition 1. For a commutative ring \( R \) and a prime number \( p \), we define the ring of Witt vectors \( W _ p(R) \) as the ring consisting of \( (X _ 0, X _ 1, \ldots) \in R^{\mathbb{N}} \) with addition and multiplication given as above.

Assume for now that this is a well-defined ring. We seem to be departing from our original example of \( \mathcal{O} _ K \) because the condition \[ \displaystyle \sum _ {n=0}^{\infty} p^n \tau(x _ n) + \sum _ {n=0}^{\infty} p^n \tau(y _ n) = \sum _ {n=0}^{\infty} p^n \tau((x+y) _ n) \] reduces mod \( p^{n+1} \) gives \[ \displaystyle \sum _ {j=0}^{n} p^j x _ j^{\lvert k \rvert^{n-j}} + \sum _ {j=0}^{n} p^j y _ j^{\lvert k \rvert^{n-j}} \equiv \sum _ {j=0}^{n} p^j (x+y) _ j^{\lvert k \rvert^{n-j}} \pmod{p^{n+1}}. \] But actually \( \alpha^{p^t \lvert k \rvert} \equiv \alpha^{p^t} \) modulo \( p^{t+1} \) and so after applying \( \mathrm{Frob}^n \) on both sides this is equivalent to \[ \displaystyle \sum _ {j=0}^{n} p^j x _ j^{p^n} + \sum _ {j=0}^{n} p^j x _ j^{p^n} \equiv \sum _ {j=0}^{n} p^j (x+y) _ j^{p^n} \pmod{p^{n+1}}. \]

Proposition 2. For \( \mathbb{F} _ q \) the finite field with \( q = p^n \) elements, the ring of Witt vectors \( W _ p(\mathbb{F} _ q) \) is isomorphic to \( \mathcal{O} _ K \) where \( K = \mathbb{Q} _ p(\zeta _ {q-1}) \) is the unique unramified extension of \( \mathbb{Q} _ p \) of degree \( n \). This isomorphism is given by \[ \displaystyle (X _ 0, X _ 1, \ldots) \mapsto \sum _ {n=0}^{\infty} p^n \tau(X _ n^{1/p^n}) \] where raising to the power of \( 1/p \) is the inverse of the Frobenius automorphism.

3. Construction of addition and multiplication#

For the construction of addition and multiplication, we shall look at a general case. If we don’t restrict to a specific prime \( p \), we may consider the following generalization.

A universal Witt vector is defined as a sequence \( (X _ 1, X _ 2, \ldots) \) and its ghost component is \[ \displaystyle X^{(n)} = \sum _ {d \mid n}^{} d X _ {d}^{n/d}. \] Again, our goal is to construct \( (X+Y) _ n \) and \( (XY) _ n \) as integer coefficient polynomials in \( X _ n \) and \( Y _ n \) so that \[ \displaystyle (X+Y)^{(n)} = X^{(n)} + Y^{(n)}, \quad (XY)^{(n)} = X^{(n)} Y^{(n)}. \] If we succeed, we can also define the ring of universal Witt vectors \( W(R) \) for an arbitrary commutative ring \( R \).

How does this have to do with the \( p \)-version of Witt vectors? Note that \( X^{(n)} \) only depends on the \( X _ d \) for \( d \) a divisor of \( n \). So if we let \( P = \{1, p, p^2, \ldots \} \), there is a natural projection map \[ \displaystyle \prod _ {n=1}^{\infty} R = \{ (X _ 1, X _ 2, \ldots) \} \rightarrow \{ (X _ 1, X _ p, X _ {p^2}, \ldots) \} = \prod _ {n \in P}^{} R, \] which will induce a surjective ring homomorphism \[ \displaystyle W(R) \twoheadrightarrow W _ p(R). \] Conversely, if we manage to show that addition and multiplication are well-defined for universal Witt vectors, we can ignore all the other components to get well-defined addition and multiplication for \( p \)-Witt vectors. That is, it suffices to construct addition and multiplication for universal Witt vectors.

Let us look at the power series \[ \displaystyle f _ X(t) = \prod _ {n=1}^{\infty} (1 - X _ n t^n). \] If we take the logarithmic derivative, we get \[ \displaystyle \frac{d}{dt} \log f _ X = \sum _ {n=1}^{\infty} \frac{-n X _ n t^{n-1}}{1 - X _ n t^n} = -\frac{1}{t} \sum _ {n=1}^{\infty} \sum _ {n \mid m}^{} n X _ n^{m/n} t^m = -\frac{1}{t} \sum _ {m=1}^{\infty} X^{(m)} t^m. \] This shows that, assuming \( X+Y \) exists with the desired property, \[ \displaystyle \frac{d}{dt} \log f _ {X+Y} = \frac{d}{dt} \log f _ X + \frac{d}{dt} \log f _ Y. \] Because all \( f _ X \), \( f _ Y \), \( f _ {X+Y} \) have constant terms \( 1 \), we further get \[ \displaystyle f _ {X+Y} = f _ X f _ Y, \] and this means that \[ \displaystyle \prod _ {n=1}^{\infty} (1 - (X+Y) _ n t^n) = \prod _ {n=1}^{\infty} (1 - X _ n t^n) (1 - Y _ n t^n). \] Comparing the \( t^n \) coefficients inductively shows that \( (X+Y) _ n \) can be expressed as a polynomial in \( X _ n \) and \( Y _ n \) with integer coefficients. Moreover, \[ \displaystyle \prod _ {n=1}^{\infty} (1 - (-X) _ n t^n) = \prod _ {n=1}^{\infty} (1 + X _ n t^n + X _ n^2 t^{2n} + \cdots) \] and so \( (-X) _ n \) also can be expressed as integer coefficient polynomials in \( X _ i \).

We use a similar argument for \( XY \). We note that \[ \displaystyle \begin{aligned} \frac{d}{dt} \log f _ {XY} & \displaystyle= -\frac{1}{t} \sum _ {n=1}^{\infty} X^{(n)} Y^{(n)} t^n = -\frac{1}{t} \sum _ {n=1}^{\infty} \sum _ {d,e \mid n}^{} d e X _ d^{n/d} Y _ e^{n/e} t^n \\ & \displaystyle= -\frac{1}{t} \sum _ {d,e=1}^{\infty} \sum _ {\mathrm{lcm}(d, e) \mid n}^{} de X _ d^{n/d} Y _ e^{n/e} t^n = \sum _ {d,e=1}^{\infty} \frac{- d e X _ d^{m/d} Y _ e^{m/e} t^{m-1}}{1 - X _ d^{m/d} Y _ e^{m/e} t^m} \\ & \displaystyle= \sum _ {d,e=1}^{\infty} \frac{de}{m} \frac{d}{dt} \log (1 - X _ d^{m/d} Y _ e^{m/e} t^m), \end{aligned} \] where \( m = \mathrm{lcm}(d, e) \). Comparing the constant terms gives \[ \displaystyle f _ {XY} = \prod _ {d,e=1}^{\infty} (1 - X _ d^{m/d} Y _ e^{m/d} t^m)^{de/m}, \] and comparing the \( t^n \) coefficients shows that \( (XY) _ n \) are polynomials in \( X _ i \) and \( Y _ i \) with integer coefficients. Therefore addition and multiplication of Witt vectors are well-defined over every ring \( R \).

We have seen how to restrict attention to \( X _ 1, X _ p, X _ {p^2}, \ldots \) and define \( W _ p(R) \). We note that this can be done more generally.

Definition 3. Let \( P \subset \mathbb{Z} _ {\ge 1} \) be a subset that is closed under divisors, i.e., \( x \in P \) and \( y \mid x \) implies \( y \in P \). Then \( X^{(n)} \) for \( n \in P \) only depends on \( X _ m \) for \( m \in P \) and hence \( (X+Y) _ n \) and \( (XY) _ n \) all depend only on \( X _ m, Y _ m \) for \( m \in P \). Thus the ring \[ \displaystyle W _ P(R) = \prod _ {n \in P}^{} R = \{ (X _ n) _ {n \in P} \} \] is well-defined and there is a natural projection map \( W(R) \twoheadrightarrow W _ P(R) \).

Example 4. We have already seen \( W _ {\{1, p, p^2, \ldots \}}(R) = W _ p(R) \). We can also define the truncated ring \[ \displaystyle W _ {p,n}(R) = W _ {\{1, p, p^2, \ldots, p^{n-1}\}}(R). \] If \( R = \mathbb{F} _ {q} \) so that \( W _ p(\mathbb{F} _ q) = \mathcal{O} _ K \), then \( W _ {p,n}(\mathbb{F} _ q) = \mathcal{O} _ K / (p^n) \).

4. Frobenius and Verschiebung#

Let \( k \) be a field of characteristic \( p \). There is a Frobenius endomorphism \[ \displaystyle \mathrm{Frob} : k \rightarrow k; \quad x \mapsto x^p. \] Because the construction of \( W \) is functorial, we get a ring homomorphism \[ \displaystyle F : W _ p(k) \rightarrow W _ p(k); \quad (X _ 0, X _ 1, \ldots) \mapsto (X _ 0^p, X _ 1^p, \ldots). \] Note that in the case \( k = \mathbb{F} _ q \), this is actually the Frobenius automorphism of \( K = \mathbb{Q} _ p(\zeta _ {q-1}) \).

There is also a shift(Verschiebung) map \[ \displaystyle V : W _ p(k) \rightarrow W _ p(k); \quad (X _ 0, X _ 1, \ldots) \mapsto (0, X _ 0, X _ 1, \ldots). \] For \( k = \mathbb{F} _ q \), this will correspond to multiplication by \( p \) and then applying inverse Frobenius. In this section, we are going to analyze these maps \( F \) and \( V \) in the generality of arbitrary fields \( k \).

Setting \( k \) to be a characteristic \( p \) field and then working over \( W _ p(k) \) can be misleading because division by \( p \) doesn’t work well over \( k \). Hence we are going to work with integer coefficient polynomials for the moment. From the definition \[ \displaystyle X^{(n)} = X _ 0^{p^n} + p X _ 1^{p^{n-1}} + \cdots + p^{n-1} X _ {n-1}^p + p^n X _ n, \] it is clear that \[ \displaystyle (VX)^{(n)} = p X^{(n-1)}, \quad (FX)^{(n-1)} = X^{(n)} - p^n X _ n. \] It immediately follows that \( V : W _ p \rightarrow W _ p \) is additive, i.e., a homomorphism of abelian groups. (Note that \( F \) is not a ring homomorphism in the general case.)

Proposition 5. If \( k \) is a field with \( \mathrm{char} k = p \), \[ \displaystyle VF = FV = p \] in \( W _ p(k) \). (This holds more generally in \( W _ p(R) \) where \( R \) is a ring with \( p = 0 \).)

Let me establish one more identity.

Proposition 6. If \( k \) is a field with \( \mathrm{char} k = p \), (more generally a ring with \( p = 0 \)) then in \( W _ p(k) \) we have \[ \displaystyle (V^i X) (V^j Y) = V^{i+j}(F^j X \cdot F^i Y). \]

Proposition 7. Let \( k \) be a field of characteristic \( p \). If \( X \in W _ p(k) \) has \( X _ 0 \neq 0 \), then \( X \) is a unit in \( W _ p(k) \).

This is a generalization of the fact that \( x \in \mathcal{O} _ K \) is a unit if \( v(x) = 0 \).

Proposition 8. Let \( k \) be a field of characteristic \( p \), such that the Frobenius is an automorphism. Then \( W _ p(k) \) is a complete discrete valuation ring with valuation \( v(X) \) being the least \( n \) with \( X _ n \neq 0 \), and residue field \( k \).

Remark 9. In the case when \( p \neq 0 \) in \( R \), the Frobenius map \( F : W _ p(R) \rightarrow W _ p(R) \) doesn’t really behave well. It is indeed possible to define \( F \) in a general situation so that \( FV = p \) over general \( R \). This Frobenius will not be just raising to a power of \( p \), but some more complicated thing. But we don’t need this complicated definition of \( F \). See [Rab14] for more on the correct Frobenius.

5. Artin–Schreier theory#

It turns out that we can do a lot of Galois theory with Witt vectors. In fact, Witt proved a generalization of Artin–Schreier theory, also called Artin–Schreier–Witt theory that applies to cyclic extensions of degree \( p^n \).

Recall that the classical Artin–Schreier theory worked in this way. Let \( K \) be a field with characteristic \( p \). Consider the map \( \wp(x) = x^p - x \), which is an additive homomorphism \( K \rightarrow K \). We have a short exact sequence \[ \displaystyle 0 \rightarrow \mathbb{F} _ p^+ \rightarrow (K^\mathrm{sep})^+ \xrightarrow{\wp} (K^\mathrm{sep})^+ \rightarrow 0 \] and taking Galois invariants gives a long exact sequence \[ \displaystyle \begin{aligned} 0 & \displaystyle\rightarrow \mathbb{F} _ p^+ \rightarrow K^+ \xrightarrow{\wp} K^+ \\ & \displaystyle\rightarrow H^1(\mathrm{Gal}(K^\mathrm{sep}/K), \mathbb{F} _ p^+) \rightarrow H^1(\mathrm{Gal}(K^\mathrm{sep}/K), (K^\mathrm{sep})^+) \rightarrow \cdots. \end{aligned} \] Here, the nontrivial fact is that \( H^1 \) of \( (K^\mathrm{sep})^+ \) vanishes and thus we get \[ \displaystyle \mathrm{Hom}(\mathrm{Gal}(K^\mathrm{sep}/K), \mathbb{F} _ p^+) \cong K^+ / \wp K^+. \] A nontrivial element of the left hand side corresponds to a cyclic extension of \( K \) of degree \( p \), and then tracing through the definitions show that this extension must be the splitting field of \( x^p - x - a \) for a nonzero \( a \in K^+ / \wp K^+ \).

Our goal is to develop an analogue of this with truncated Witt vectors. The idea is that \( K = W _ {p,1}(K) \) and so if we use higher truncations, we might be able to recover more information. Let \( K \) be a field with characteristic \( p \). First note that for \( \sigma \in \mathrm{Gal}(K^\mathrm{sep}/K) \), the map \[ \displaystyle \sigma : W _ {p,n}(K^\mathrm{sep}) \rightarrow W _ {p,n}(K^\mathrm{sep}); \quad (X _ 0, \ldots, X _ {n-1}) \mapsto (\sigma X _ 0, \ldots, \sigma X _ {n-1}) \] is a ring automorphism, because \( \sigma \) acts trivially on the integer coefficients. Thus \( \mathrm{Gal}(K^\mathrm{sep}/k) \) acts on \( W _ {p,n}(K^\mathrm{sep}) \) in a canonical way.

Next, consider the map \[ \displaystyle \wp : W _ {p,n}(K^\mathrm{sep}) \rightarrow W _ {p,n}(K^\mathrm{sep}); \quad X \mapsto FX - X. \] Because \( F \) is additive, this map \( \wp \) is also additive. Moreover, it restricts to \( W _ {p,n}(K) \rightarrow W _ {p,n}(K) \). The kernel is going to be the set of \( X \) such that \( FX = X \), and hence \( W _ {p,n}(\mathbb{F} _ p) \). That is, \[ \displaystyle 0 \rightarrow W _ {p,n}(\mathbb{F} _ p)^+ \rightarrow W _ {p,n}(K^\mathrm{sep})^+ \xrightarrow{\wp} W _ {p,n}(K^\mathrm{sep})^+ \] is exact. Here note that \( W _ {p,n}(\mathbb{F} _ p)^+ \cong \mathbb{Z} _ p / (p^n) \cong \mathbb{Z}/p^n \mathbb{Z} \) is the cyclic group.

Proposition 10. The map \( \wp : W _ {p,n}(K^\mathrm{sep})^+ \rightarrow W _ {p,n}(K^\mathrm{sep})^+ \) is surjective.

So we have \[ \displaystyle 0 \rightarrow W _ {p,n}(\mathbb{F} _ p)^+ \rightarrow W _ {p,n}(K^\mathrm{sep})^+ \xrightarrow{\wp} W _ {p,n}(K^\mathrm{sep})^+ \rightarrow 0 \] and taking \( \mathrm{Gal}(K^\mathrm{sep}/K) \)-invariants gives \[ \displaystyle \begin{aligned} 0 & \displaystyle\rightarrow W _ {p,n}(\mathbb{F} _ p)^+ \rightarrow W _ {p,n}(K)^+ \xrightarrow{\wp} W _ {p,n}(K)^+ \\ & \displaystyle\rightarrow \mathrm{Hom}(\mathrm{Gal}(K^\mathrm{sep}/K), W _ {p,n}(\mathbb{F} _ p)^+) \rightarrow H^1(\mathrm{Gal}(K^\mathrm{sep}/K), W _ {p,n}(K^\mathrm{sep})^+) \rightarrow \cdots. \end{aligned} \]

6. Vanishing of \( H^1 \)#

We now show that as in the classical Artin–Schreier theory, the \( H^1 \) term vanishes.

Proposition 11. \( H^1(\mathrm{Gal}(K^\mathrm{sep}/K), W _ {p,n}(K^\mathrm{sep})^+) = 0 \).

Now we get an exact sequence \[ \displaystyle 0 \rightarrow W _ {p,n}(\mathbb{F} _ p)^+ \rightarrow W _ {p,n}(K)^+ \xrightarrow{\wp} W _ {p,n}(K)^+ \xrightarrow{\delta} \mathrm{Hom}(\mathrm{Gal}(K^\mathrm{sep}/K), W _ {p,n}(\mathbb{F} _ p)^+) \rightarrow 0. \] Recall that \( W _ {p,n}(\mathbb{F} _ p)^+ \) is cyclic of order \( p^n \). So cyclic extensions of \( K \) of degree \( p^n \) correspond to continuous surjective homomorphisms \[ \displaystyle \mathrm{Gal}(K^\mathrm{sep}/K) \rightarrow W _ {p,n}(\mathbb{F} _ p)^+. \]

But the boundary map \( \delta \) induces an isomorphism \[ \displaystyle \delta : W _ {p,n}(K)^+ / \wp W _ {p,n}(K)^+ \xrightarrow{\cong} \mathrm{Hom}(\mathrm{Gal}(K^\mathrm{sep}/K), W _ {p,n}(\mathbb{F} _ p)^+) \] given by \[ \displaystyle \delta : [X] \mapsto (\sigma \mapsto \sigma(Y) - Y) \] where \( Y \in W _ {p,n}(K^\mathrm{sep}) \) satisfies \( \wp(Y) = X \). If \( L / K \) is a cyclic extension of order \( p^n \), and corresponds to some continuous surjective homomorphism \( \Phi : \mathrm{Gal}(K^\mathrm{sep} / K) \rightarrow W _ {p,n}(\mathbb{F} _ p)^+ \), its kernel will be \( \ker(\Phi) = \mathrm{Gal}(K^\mathrm{sep}/L) \). But \[ \displaystyle \ker(\sigma \mapsto \sigma(Y) - Y) = \{ \sigma : \sigma(Y) = Y \} = \mathrm{Gal}(K^\mathrm{sep} / K(Y _ 0, \ldots, Y _ {n-1})). \] Therefore \( L = K(Y _ 0, \ldots, Y _ {n-1}) \).

Theorem 12 (Witt). Let \( K \) be a field of characteristic \( p \). Any cyclic extension \( L / K \) of degree \( p^n \) is of the form \[ \displaystyle L = K(Y _ 0, Y _ 1, \ldots, Y _ {n-1}) \] where \( Y \in W _ {p,n}(K^\mathrm{sep}) \) satisfies \( FY - Y = X \) for some \( X \in W _ {p,n}(K) \). In other words, \( L \) is the splitting field of the system of equations \[ \displaystyle Y _ 0^p = (X+Y) _ 0, \quad Y _ 1^p = (X+Y) _ 1, \quad \cdots, \quad Y _ {n-1}^p = (X+Y) _ {n-1} \] for some \( X _ 0, \ldots, X _ {n-1} \in K \).

References#

[Lan02] Serge Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556

[Rab14] Joseph Rabinoff, The theory of Witt vectors, arXiv:1409.7445 (2014).

[Wit37] Ernst Witt, Zyklische Kouml;rper und Algebren der Charakteristik p vom Grad pn. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p, J. Reine Angew. Math. 176 (1937), 126–140. MR 1581526