2. Discussion of the axioms

Let K be an arbitrary field and p a prime number. We briefly recall some basic properties of the Brauer group, and define the pairing function

\[ \text{Hom}(\text{G}_K, \mathbb{Z}/p) \times K^\times/p \to \text{Br}(K)[p] \]

occurring in our axioms. Here and throughout we write $\text{Hom}(\text{G}_K, \mathbb{Z}/p)$ for the (discrete) abelian group of continuous group homomorphisms from $\text{G}_K$ to the discrete group $\mathbb{Z}/p$ , we write $K^\times/p$ for the group $K^\times/K^{\times p}$ of pth power classes, and we write $\text{Br}(K)[p]$ for the set of elements of $\text{Br}(K)$ of order dividing p.

The Brauer group $\text{Br}(K)$ of K can equivalently be defined as the Galois cohomology group $H^2(\text{G}_K, {K^{\mathrm{sep}}}^\times)$ , or as the collection of central simple algebras over K modulo Brauer equivalence with the tensor product as a binary operation. We will not need this second viewpoint, which is explored in, for instance, [Reference Gille and SzamuelyGS17, § 2.4]; see [Reference Gille and SzamuelyGS17, Theorem 4.4.3] for the equivalence of the definitions.

Given a character $\chi \colon \text{G}_K \to \mathbb{Z}/p$ and an element $a \in K^\times$ , we can define an element $(\chi, a) \in \text{Br}(K)$ as follows (see [Reference SerreSer68, Chapitre XIV, § 1] or [Reference Gille and SzamuelyGS17, Proposition 4.7.3]). Observe that $\chi \in \text{Hom}(\text{G}_K, \mathbb{Z}/p) = H^1(\text{G}_K, \mathbb{Z}/p)$ , and we have an exact sequence $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/p \to 0$ of abelian groups with trivial $\text{G}_K$ -action, where the first map is multiplication by p. Hence we can apply the associated boundary map $\delta$ in cohomology to obtain an element $\delta(\chi) \in H^2(\text{G}_K, \mathbb{Z})$ . Secondly, we have $a \in K^\times = H^0(\text{G}_K, {K^{\mathrm{sep}}}^\times)$ . Taking the cup product of these two cohomology classes we obtain $(\chi, a) := \delta(\chi) \cup a \in H^2(\text{G}_K, {K^{\mathrm{sep}}}^\times) = \text{Br}(K)$ . From the central simple algebra viewpoint, $(\chi, a)$ is the class of the cyclic algebra associated to $\chi$ and a. By construction, this gives a bilinear map of abelian groups $\text{Hom}(\text{G}_K, \mathbb{Z}/p) \times K^\times \to \text{Br}(K)$ . Since $\text{Hom}(\text{G}_K, \mathbb{Z}/p)$ is a p-torsion group, it follows that the image must actually lie in $\text{Br}(K)[p]$ , and the second argument factors through $K^\times/p$ . Thus we have the desired bilinear pairing $\text{Hom}(\text{G}_K, \mathbb{Z}/p) \times K^\times/p \to \text{Br}(K)[p]$ . This pairing induces a group homomorphism

\[ K^\times \to \text{Hom}(\text{Hom}(\text{G}_K, \mathbb{Z}/p), \text{Br}(K)[p]),\]

where the right-hand side is the group of homomorphisms $\text{Hom}(\text{G}_K, \mathbb{Z}/p) \to \text{Br}(K)[p]$ of discrete groups.

An essential property of the pairing $(\cdot, \cdot)$ is given in the following lemma.

Let us show that local fields $\mathbb{F}_q(\!(t)\!)$ satisfy the axioms of Theorem 1.1.

Proposition 2.2. Suppose K is a non-archimedean local field of arbitrary characteristic; for instance, $K = \mathbb{F}_q(\!(t)\!)$ for some prime power q (not necessarily a power of p). Then $\text{Br}(K)[p] \cong \mathbb{Z}/p$ , and the pairing above induces an isomorphism

\[ K^\times/p \cong \text{Hom}(\text{Hom}(\text{G}_K, \mathbb{Z}/p), \text{Br}(K)[p]). \]

Secondly, class field theory provides the reciprocity homomorphism

\[ (\cdot, K) \colon K^\times \to \text{G}_K/[\text{G}_K, \text{G}_K] = \text{G}_K^{\mathrm{ab}},\]

which fits into the following exact sequence of abelian groups [Reference Neukirch, Schmidt and WingbergNSW08, Theorem 7.2.11] (cf. also [Reference SerreSer68, Chapitre XIII, § 4, Remarque after Proposition 13]):

\[ 0 \to K^\times \to \text{G}_K^{\text{ab}} \to \hat{\mathbb{Z}}/\mathbb{Z} \to 0. \]

Since the group $\hat{\mathbb{Z}}/\mathbb{Z}$ is torsion-free and divisible, a simple diagram chase shows that the induced homomorphism $K^\times/p \to \text{G}_K^{\mathrm{ab}}/p$ is an isomorphism. (Alternatively, take the tensor product of the sequence above with the abelian group $\mathbb{Z}/p$ , and use the fact that $\operatorname{Tor}^\mathbb{Z}_1(\hat{\mathbb{Z}}/\mathbb{Z}, \mathbb{Z}/p)$ and $(\hat{\mathbb{Z}}/\mathbb{Z})/p$ both vanish, again since $\hat{\mathbb{Z}}/\mathbb{Z}$ is torsion-free and divisible.) By Pontryagin duality, we further have a canonical isomorphism $\text{G}_K^{\mathrm{ab}}/p \cong \text{Hom}(\text{Hom}(\text{G}_K^{\mathrm{ab}}/p, \mathbb{Z}/p), \mathbb{Z}/p) = \text{Hom}(\text{Hom}(G_K, \mathbb{Z}/p), \mathbb{Z}/p)$ .

We obtain a diagram

where the arrow on the left is induced by the pairing, the arrow at the top is induced by the reciprocity map and Pontryagin duality, and the diagonal arrow is induced by the canonical isomorphism $\operatorname{inv}_K$ . The diagram commutes; see [Reference SerreSer68, Chapitre XIV, § 1, Proposition 3] or [Reference Neukirch, Schmidt and WingbergNSW08, Proposition 7.2.12]. Since the top and the diagonal arrows are isomorphisms, the same must be true for the arrow on the left.

A basic observation concerning fields satisfying the axioms of Theorem 1.1 is the following.

The hypothesis on p-cohomological dimension is satisfied by the absolute Galois group $\text{G}_{\mathbb{F}_q(\!(t)\!)}$ , as indeed it is by the absolute Galois group of any field of characteristic p [Reference Neukirch, Schmidt and WingbergNSW08, Corollary 6.1.3]. We later give a strengthened version of Lemma 2.3, only requiring Galois-theoretic information on a certain quotient of $\text{G}_K$ . Since this is more complicated to prove, and is not needed for Theorem 1.1, we defer it to Lemma 5.1.

Let us now discuss the rephrasing of Theorem 1.1 as Theorem 1.2, using Kato’s cohomology groups $H^i(K, \mathbb{Z}/p(j))$ . For K of characteristic not p, these groups are defined as Galois cohomology groups $H^i(\text{G}_K, \mu_p^{\otimes j})$ and, in particular, $H^2(K, \mathbb{Z}/p(1)) \cong \text{Br}(K)[p]$ [Reference KatoKat86, p.143]. By Lemma 2.3, we may therefore assume for the purposes of comparing the axioms of Theorems 1.1 and 1.2 that K has characteristic p.

We may then take it as the definition of these groups that $H^i(K, \mathbb{Z}/p(i)) = \operatorname{K}^M_i(K)/p$ and $H^{i+1}(K, \mathbb{Z}/p(i)) = H^1(\text{G}_K, \operatorname{K}^M_i(K^{\mathrm{sep}})/p)$ , where the last group is a Galois cohomology group, and $\operatorname{K}^M_\ast$ denotes the graded Milnor K-ring. (See [Reference MerkurjevMer03, Appendix A, p.152] and the Bloch–Kato–Gabber Theorem, cf. [Reference KatoKat86, p.149, (1.2.1)].)

This means we have $H^1(K, \mathbb{Z}/p(1)) = \operatorname{K}^M_1(K)/p = K^\times/p$ , $H^1(K, \mathbb{Z}/p(0)) = H^1(\text{G}_K, \mathbb{Z}/p) = \text{Hom}(\text{G}_K, \mathbb{Z}/p)$ , and $H^2(K, \mathbb{Z}/p(1)) = H^1(\text{G}_K, {K^{\mathrm{sep}}}^\times/p) \cong \text{Br}(K)[p]$ (canonically) [Reference MerkurjevMer03, Appendix A, Example A.3(2)]. Since $H^1(K, \mathbb{Z}/p(1)) = K^\times/p$ embeds into the Galois cohomology group $H^0(\text{G}_K, {K^{\mathrm{sep}}}^\times/p)$ , the cup product in Galois cohomology yields a bilinear pairing

\[ H^1(K, \mathbb{Z}/p(0)) \times H^1(K, \mathbb{Z}/p(1)) \to H^2(K, \mathbb{Z}/p(1)) ;\]

cf. [Reference MerkurjevMer03, Appendix A, (A.5)]. Standard facts (namely the compatibility of the cup product with the connecting homomorphisms in long exact sequences) show that this pairing agrees, up to sign, with the pairing into $\text{Br}(K)[p]$ defined above. This gives the equivalence of axioms (Pair) and (Pairʹ).

To establish the equivalence of Theorems 1.1 and 1.2, we also need to translate the axiom that $H^2(K, \mathbb{Z}/p(2)) = \operatorname{K}^M_2(K)/p$ vanishes into the statement that $[K : K^p] \leq p$ . Consider two elements $a, b \in K^\times$ . These are p-dependent, meaning that $[K^p(a,b) : K^p] \leq p$ if and only if the elements da and db of the vector space of absolute Kähler differentials $\Omega_K$ are linearly dependent, which is the case if and only if the differential form $da \wedge db \in \Omega^2_K$ vanishes; cf. [Reference Gille and SzamuelyGS17, Proposition A.8.11] or [Reference BourbakiBou07, Chapitre V, No 2, § 13, Théorème 1 a)]. By the Bloch–Kato–Gabber Theorem [Reference Gille and SzamuelyGS17, Theorem 9.5.2], the group $\operatorname{K}^M_2(K)/p$ naturally embeds into $\Omega^2_K$ , from which we deduce that $da \wedge db \in \Omega^2_K$ vanishes if and only if the symbol $\{ a, b \} \in \operatorname{K}^M_2(K)/p$ vanishes. This shows that the group $\operatorname{K}^M_2(K)/p$ vanishes if and only if any two $a, b \in K^\times$ are p-dependent, i.e. if and only if $[K : K^p] \leq p$ .

3. Existence of a valuation

Let K be a field with $\text{G}_K \cong \text{G}_{\mathbb{F}_q(\!(t)\!)}$ . The goal of this section is to show that K carries a non-trivial henselian valuation v whose residue field Kv is not p-closed, i.e. such that Kv has a $\mathbb{Z}/p$ -extension, or equivalently the maximal pro-p quotient $\text{G}_{Kv}(p)$ is not trivial.

To see that this is false, consider the profinite group $H = \mathbb{Z}_2 \ltimes \prod_{l \neq 2} \mathbb{Z}_l$ , where the product is over odd primes l, and $x \in \mathbb{Z}_2$ acts on $\prod_{l \neq 2} \mathbb{Z}_l$ as multiplication by $(-1)^x$ . Then every Sylow subgroup of H is procyclic. However, for $p \neq 2$ , the maximal prime-to-p quotient H(p’) is the non-abelian $\mathbb{Z}_2 \ltimes \prod_{l \neq 2, p} \mathbb{Z}_l$ , which contradicts [Reference Efrat and FesenkoEF99, Lemma 4.2]. One can also check that the maximal prime-to-2 quotient H(2’) is the trivial group, for a further contradiction. The mistake in the proof of [Reference Efrat and FesenkoEF99, Lemma 4.2] lies in an erroneous statement concerning the canonical projection $\hat{F} \to \hat{F}(\mathrm{ab}, p')$ in the notation there, where $\hat{F}$ is a free profinite group.

Although parts of the proof of [Reference Efrat and FesenkoEF99, Proposition 4.3] could be salvaged, we use this opportunity to give a new full proof of [Reference Efrat and FesenkoEF99, Proposition 4.3], or rather of a slightly strengthened formulation.

(We mention in passing that this also follows very neatly from Koenigsmann’s characterisation of ‘tamely branching valuations’ as presented in [Reference Engler and PrestelEP05, Theorem 5.4.3]. Indeed, by the theorem the canonical henselian valuation v on K must be tamely branching at all $l \neq p$ since the same is true for $\mathbb{F}_q(\!(t)\!)$ and this is a Galois-theoretic condition; the residue field Kv must have characteristic p by the argument of [Reference Efrat and FesenkoEF99, Theorem 4.1(b)]; and since the abelian normal subgroups of l-Sylow subgroups of $\text{G}_{\mathbb{F}_q(\!(t)\!)}$ are procyclic, it follows that $[vK : l\cdot vK] = l$ from ramification theory.)

It remains to prove the assertions about $\text{G}_{Kv}(p')$ and $Kv \cap \overline{\mathbb{F}_p}$ . We extensively use the theory of ramification and inertia subgroups as presented, for instance, in [Reference EfratEfr06, Chapter 15]. Let $\sigma \colon \text{G}_{\mathbb{F}_q(\!(t)\!)} \to \text{G}_K$ be an isomorphism, and write T, V for the inertia and ramification subgroups of $\text{G}_{\mathbb{F}_q(\!(t)\!)}$ . Let $L/K$ be the fixed field of $\sigma(V) \leq \text{G}_K$ . Then $L/K$ is a Galois extension with Galois group isomorphic to $\text{G}_{\mathbb{F}_q(\!(t)\!)}/V$ , so it is generated as a profinite group by two elements $\sigma$ , $\tau$ with the sole relation $\sigma\tau\sigma^{-1} = \tau^q$ , i.e. $\text{Gal}(L/K)$ is the semi-direct product of $\langle\sigma\rangle \cong \hat{\mathbb{Z}}$ and $\langle\tau\rangle \cong \hat{\mathbb{Z}}/\mathbb{Z}_p$ [Reference Neukirch, Schmidt and WingbergNSW08, Theorem 7.5.3]. Furthermore, the absolute Galois group of L is isomorphic to V, and is thus a free pro-p group. In particular, for every prime number $l \neq p$ the value group vL (where we also write v for the unique prolongation of the valuation v on K to L) is l-divisible.

The relative ramification subgroup R of $\text{Gal}(L/K)$ is a normal pro-p subgroup. The p-Sylow subgroups of $\text{Gal}(L/K) = \langle \sigma, \tau\rangle$ are procyclic, so R is abelian. By [Reference Neukirch, Schmidt and WingbergNSW08, Corollary 7.5.7(ii)], R must be contained in $\langle\tau\rangle \cong \hat{\mathbb{Z}}/\mathbb{Z}_p$ , which has no pro-p part. Therefore R is trivial, i.e. the extension $L/K$ is tamely ramified.

Thus the relative inertia group I of $L/K$ is an abelian normal subgroup of $\text{Gal}(L/K) = \langle \sigma, \tau \rangle$ . By [Reference Neukirch, Schmidt and WingbergNSW08, Corollary 7.5.7(ii)] once more, $I \leq \langle \tau\rangle \cong \hat{\mathbb{Z}} / \mathbb{Z}_p$ . We have the so-called ramification pairing

\[ I \times vL/vK \to \mu(Lv), \]

where $\mu(Lv)$ is the group of roots of unity in the residue field Lv (see [Reference EfratEfr06, Theorem 16.2.6]). (In the notation there, observe that L is the ramification subfield of $L/K$ with respect to v since $L/K$ is tamely ramified, and the value group vK agrees with the value group of the relative inertia field since $L/K$ is separable, cf. the diagram on [Reference EfratEfr06, p.149].) The ramification pairing is bilinear with trivial left and right kernel, and it is continuous (where $vL/vK$ and $\mu(Lv)$ carry the discrete topology and I its profinite topology). Since vL is l-divisible for every prime number $l \neq p$ , but vK is not, the torsion group $vL/vK$ contains elements of arbitrary l-power order. The triviality of the right kernel of the pairing then implies that $\mu(Lv)$ contains elements of arbitrary l-power order, and furthermore that the profinite group I must have non-trivial l-Sylow subgroup. It follows that Sylow subgroups of the quotient $\langle\tau\rangle/I$ are torsion, since the l-Sylow subgroup of both $\langle\tau\rangle$ and I is isomorphic to $\mathbb{Z}_l$ for every $l \neq p$ .

We have $\text{Gal}(Lv/Kv) \cong \langle\sigma,\tau\rangle/I$ . In particular the p-Sylow subgroups of $\text{Gal}(Lv/Kv)$ are isomorphic to $\mathbb{Z}_p$ , so $\text{cd}_p \text{Gal}(Lv/Kv) = 1$ . Since $\text{G}_{Lv}$ is a pro-p group as a quotient of $\text{G}_L$ , it follows from [Reference Neukirch, Schmidt and WingbergNSW08, Theorem 3.5.6] that the exact sequence

\[ 1 \to \text{G}_{Lv} \to \text{G}_{Kv} \to \text{Gal}(Lv/Kv) \to 1, \]

splits. Hence $\text{Gal}(Lv/Kv)$ is isomorphic to a subgroup of $\text{G}_{Kv}$ and therefore torsion-free. Therefore the subgroup $\langle\tau\rangle/I$ of $\langle\sigma,\tau\rangle/I \cong \text{Gal}(Lv/Kv)$ is also torsion-free. On the other hand, we have seen above that its Sylow subgroups are torsion, so we must have $I = \langle\tau\rangle$ . Thus $\text{Gal}(Lv/Kv) \cong \langle\sigma,\tau\rangle/I \cong \hat{\mathbb{Z}}$ . Since $\text{G}_{Lv}$ is pro-p, we deduce that $\text{G}_{Kv}(p') = \text{Gal}(Lv/Kv)(p') = \hat{\mathbb{Z}}/\mathbb{Z}_p$ .

It remains to prove that $Kv \cap \overline{\mathbb{F}_p} \cong \mathbb{F}_q$ . We saw above in our analysis of $\mu(Lv)$ that Lv contains $l^s$ -roots of unity for every prime number $l \neq p$ and every $s \geq 1$ . In other words, Lv contains the algebraic closure $\overline{\mathbb{F}_p}$ .

The ramification pairing induces an isomorphism $I \cong \text{Hom}(vL/vK, \mu(Lv))$ [Reference EfratEfr06, Corollary 16.2.7(c)], where the homomorphism group is endowed with the topology of pointwise convergence. This isomorphism is compatible with the action of $\langle\sigma\rangle \cong \text{Gal}(Lv/Kv)$ , where $\text{Gal}(Lv/Kv)$ acts on the homomorphism group via its action on $\mu(Lv)$ , and $\langle\sigma\rangle$ acts on I by conjugation.

The automorphism group $\operatorname{Aut}(\overline{\mathbb{F}_p}) = \text{Gal}(\overline{\mathbb{F}_p}/\mathbb{F}_p)$ is isomorphic to $\hat{\mathbb{Z}}$ and is generated by the Frobenius automorphism $x \mapsto x^p$ . Say the image of $\sigma$ in $\text{Gal}(Lv/Kv)$ acts on $\overline{\mathbb{F}_p}$ as $x \mapsto x^{p^n}$ for some $n \in \hat{\mathbb{Z}}$ . Therefore this element of $\text{Gal}(Lv/Kv)$ acts on the abelian profinite group $\text{Hom}(vL/vK, \mu(Lv))$ as multiplication by $p^n$ . Compatibility of the action with the isomorphism induced by the ramification pairing means that $\sigma$ must act in the same way on I by conjugation. In other words, we have $\sigma\eta\sigma^{-1} = \eta^{p^n}$ for each $\eta \in I = \langle\tau\rangle$ . However, by construction of $\langle\sigma,\tau\rangle$ as a semi-direct product we know that $\sigma\eta\sigma^{-1} = \eta^q$ . This is only possible for all elements $\eta \in I = \langle\tau\rangle \cong \hat{\mathbb{Z}}/\mathbb{Z}_p$ if we have $p^n = q$ as elements of $(\hat{\mathbb{Z}}/\mathbb{Z}_p)^\times$ . It follows that an element $x \in \overline{\mathbb{F}_p}$ lies in Kv, i.e. is fixed by $\text{Gal}(Lv/Kv)$ , if and only if $x^{p^n} = x^q = x$ . Therefore $\overline{\mathbb{F}_p} \cap Kv = \mathbb{F}_q$ .

We later give an alternative version of Corollary 3.3, requiring only weaker Galois-theoretic information, but also only yielding a p-henselian valuation. Since this is not needed for the basic result (Theorem 1.1), and the proof uses different techniques as it cannot rely on [Reference Efrat and FesenkoEF99, Theorem 4.1], we defer this to Proposition 5.3.

4. Consequences of the Brauer group axioms

Throughout this section, let K be a field such that:

1. (1) K has countably many $\mathbb{Z}/p$ -extensions;

2. (2) K has characteristic p; and

3. (3) the axioms (Brau) and (Pair) from Theorem 1.1 are satisfied.

Artin–Schreier theory yields an isomorphism $\text{Hom}(\text{G}_K, \mathbb{Z}/p) \cong K/\wp(K)$ , where $\wp(K)$ is the image of the group homomorphism $\wp \colon K \to K, x \mapsto x^p - x$ [Reference Neukirch, Schmidt and WingbergNSW08, Corollary 6.1.2]. Our first assumption can therefore also be phrased as the assertion that the group $K/\wp(K)$ is countable. The Artin–Schreier isomorphism yields a pairing $[{\cdot}, {\cdot}) \colon K/\wp(K) \times K^\times/p \to \text{Br}(K)[p]$ from the pairing defined in § 2. (This notation is also used in [Reference SerreSer68, Chapitre XIV, § 5].) We will also consider $[{\cdot}, {\cdot})$ as a function defined on $K \times K^\times$ .

In this section we show that certain valuations v on K satisfy very strong properties. In the proof of Theorem 1.1, we will apply this to a valuation obtained (under the hypothesis (Gal), beyond the standing assumptions of this section introduced above) from the results of § 3. However, for the time being we develop the material for arbitrary valuations v, of which we will generally only assume that they are non-trivial, have non-p-closed residue field Kv, and are p-henselian. This last standard condition, see [Reference Chatzidakis and PereraCP17], means that the maximal ideal $\mathfrak{m}_v$ of the valuation ring is contained in $\wp(K)$ , or equivalently that the valuation extends uniquely to every $\mathbb{Z}/p$ -extension of K. (We will meet the general condition of l-henselianity, where l is a prime not necessarily equal to the characteristic, in § 5 below.)

\[ Kv/(Kv)^p \to K/(\wp(K) + x^p \mathfrak{m}_v), (y + \mathfrak{m}_v) + (Kv)^p \mapsto yx^p + \wp(K) + x^p\mathfrak{m}_v, \]

(where $y \in \mathcal{O}_v$ ). This map is well defined since if $y \in z^p + \mathfrak{m}_v$ for some $z \in \mathcal{O}_v$ , we have

\[ yx^p \in z^px^p + x^p \mathfrak{m}_v = \wp(zx) + x^p \frac{z}{x^{p-1}} + x^p \mathfrak{m}_v \subseteq \wp(K) + x^p \mathfrak{m}_v .\]

This map is also injective. Indeed, if $y \in \mathcal{O}_v$ is such that $yx^p \in \wp(K) + x^p \mathfrak{m}_v$ , we have $y \in \mathfrak{m}_v + x^{-p}\wp(K)$ , so there exists $z \in K$ with $y - x^{-p}(z^p-z) \in \mathfrak{m}_v$ . Comparing valuations, we must have $v(z) \geq v(x)$ , and so $x^{-p}z \in \mathfrak{m}_v$ , yielding $y - (z/x)^p \in \mathfrak{m}_v$ . This shows that $y + \mathfrak{m}_v \in Kv$ is a pth power, proving that the map has trivial kernel. Since the codomain of the map is a quotient of of $K/\wp(K)$ and therefore countable by our standing assumption, the positive-dimensional $(Kv)^p$ -vector space $Kv/(Kv)^p$ must also be countable. This shows that $(Kv)^p$ and therefore Kv are countable.

Let $a \in \mathcal{O}_v$ such that the residue of a in Kv is not a pth power. For every $\gamma \in vK$ , $\gamma < 0$ , choose an element $x_\gamma$ of valuation $\gamma$ , and set $y_\gamma = a x_\gamma^p$ . We claim that the images of the $y_\gamma$ in $K/\wp(K)$ are $\mathbb{F}_p$ -linearly independent. Indeed, if they were not, then there would exist $\gamma_1 < \gamma_2 < \dotsb \gamma_n < 0$ and coefficients $a_2, \dotsc, a_n \in \mathbb{F}_p$ , as well as an element $z \in K$ with $y_{\gamma_1} = a_2 y_{\gamma_2} + \dotsb + a_n y_{\gamma_n} + z^p - z$ . Considering the valuation of each term, we must have $v(z) = v(y_{\gamma_1})/p = v(x_{\gamma_1})$ . Dividing by $x_{\gamma_1}^p$ and taking residues, we obtain $\overline{a} = \overline{z/x_{\gamma_1}}^p$ , contradicting the choice of a. This shows the linear independence claimed.

In particular, it follows that vK is countable since $K/\wp(K)$ is countable. Assume now that vK does not have a minimal positive element, so vK is densely ordered. We may then fix a sequence $\delta_1, \delta_2, \dotsc$ of negative elements of vK converging to 0. There exists a family of functions $f_i \in \mathbb{N} \to \{0, 1\}$ , indexed by i in some uncountable index set I, such that for $i \neq j$ the functions $f_i$ and $f_j$ differ in infinitely many places. (For instance, for every set P of prime numbers, choose a function f which is 1 at each power of a prime in P and 0 everywhere else.) By the pairing axiom (Pair), and using the linear independence previously proved, there exist elements $z_i \in K^\times$ , indexed by $i \in I$ , such that $[{y_{\delta_n}}, {z_i})$ is a non-zero element of $\text{Br}(K)[p]$ if and only if $f_i(n) = 1$ .

The group $K^\times/(K^{\times p}(1+\mathfrak{m}_v))$ is countable as a group extension of $K^\times/(K^{\times p} \mathcal{O}_v^\times) \cong vK/p$ by $(K^{\times p} \mathcal{O}_v^\times)/(K^{\times p}(1+\mathfrak{m}_v)) \cong \mathcal{O}_v^\times/(\mathcal{O}_v^{\times p}(1+\mathfrak{m}_v)) \cong (Kv)^\times/p$ , both of which are countable groups by the previous steps. Therefore there must exist two indices $i \neq j$ such that $z_i/z_j \in K^{\times p} (1 + \mathfrak{m}_v)$ , so we may write $z_i/z_j = a^p (1 + b)$ for suitable $a \in K^\times$ , $b \in \mathfrak{m}_v$ .

Since the $\delta_n$ converge to 0, there exists an $n_0$ such that for all $n \geq n_0$ we have $\delta_n > -v(b)$ , and therefore

\[ \left[{x_{\delta_n}}, {z_i/z_j}\right) = \left[{x_{\delta_n}}, {1+b}\right) = 0,\]

by Lemma 4.1 (set $x = x_{\delta_n}^{-1}$ there). This forces $f_i(n) = f_j(n)$ for all $n \geq n_0$ , contradicting our choice of functions $f_i$ .

Suppose now that vK has no minimal positive element, so that vK is densely ordered. Since vK has no non-trivial p-divisible convex subgroup, every open interval in vK containing 0 contains elements which are not p-divisible. We can therefore choose a sequence $\delta_1, \delta_2, \dotsc$ of negative elements of vK which converges to 0 and such that no $\delta_i$ is divisible by p.

For every n, let $x_n \in K$ with $vx_n = \delta_n$ . As above, the elements $x_n + \mathcal{O}_v + \wp(K) \in K/(\mathcal{O}_v + \wp(K))$ are $\mathbb{F}_p$ -linearly independent. Consequently there exists an $\mathbb{F}_p$ -linear map $f \colon K/\wp(K) \to \text{Br}(K)[p]$ with $f(\mathcal{O}_v) = 0$ and $f(x_n) \neq 0$ for each n. By the pairing axiom (Pair), there exists $z \in K^\times$ with $\left[{\cdot}, {z}\right) = f$ , so that $\left[{\mathcal{O}_v}, {z}\right) = 0$ and $\left[{x_n}, {z}\right) \neq 0$ for each n.

By assumption Kv is not p-closed, so there exists an element $a \in \mathcal{O}_v$ whose residue does not lie in $\wp(Kv)$ . This means that $a \not\in \wp(K)$ , and so adjoining a root of $X^p-X-a$ to K yields a $\mathbb{Z}/p$ -extension $L/K$ by Artin–Schreier theory. The valuation v extends uniquely to L by p-henselianity, and the residue field of L is a proper extension of the residue field Kv, since it contains an Artin–Schreier root of the residue of a, but by assumption Kv does not. Therefore the value group does not change in the degree p extension $L/K$ by [Reference Engler and PrestelEP05, Theorem 3.3.3]. Since $\left[{\mathcal{O}_v}, {z}\right) = 0$ , Lemma 2.1 (applied to the homomorphism $\chi \colon \text{G}_K \to \mathbb{Z}/p$ corresponding to a under the Artin–Schreier isomorphism $\text{Hom}(\text{G}_K,\mathbb{Z}/p) \cong K/\wp(K)$ , whose kernel is $\text{G}_L$ ) implies that z is a norm of the extension $L/K$ . It now follows from [Reference Engler and PrestelEP05, Remark 3.2.17] that the valuation of z must be divisible by p. Hence we may as well suppose that $z \in \mathcal{O}_v^\times$ by multiplying z by a pth power.

Since Kv is perfect, we may further suppose that $z \in 1 + \mathfrak{m}_v$ . However, Lemma 4.1 now implies that for all n with $v(x_n) > -v(z-1)$ we have $\left[{x_n}, {z}\right) = 0$ ; since the values $v(x_n)$ converge to 0, this is the case for all sufficiently large n, contradicting the choice of z.

As usual, an element of a valued field whose valuation is minimal positive is called a uniformiser.

Proof. By Lemma 4.2 and Lemma 4.3 the value group vK is countable. The embedding $({1}/{\pi})\mathcal{O}_v \hookrightarrow K$ induces an injection $({1}/{\pi})\mathcal{O}_v/\wp(\mathcal{O}_v) \hookrightarrow K/\wp(K)$ of $\mathbb{F}_p$ -vector spaces, which in turn induces a surjective linear map $\text{Hom}(K/\wp(K), \text{Br}(K)[p]) \to \text{Hom}(({1}/{\pi})\mathcal{O}_v/\wp(\mathcal{O}_v), \text{Br}(K)[p])$ . We therefore obtain a surjective group homomorphism

\[ K^\times/p \to \text{Hom}(K/\wp(K), \text{Br}(K)[p]) \to \text{Hom}\Bigl(\frac{1}{\pi}\mathcal{O}_v/\wp(\mathcal{O}_v), \text{Br}(K)[p]\Bigr), \]

where the first map is induced by the pairing function and is surjective by the pairing axiom (Pair). The group $1 + \pi^2\mathcal{O}_v/(K^{\times p} \cap (1 + \pi^2\mathcal{O}_v))$ lies in the kernel by Lemma 4.1, so we obtain a surjective homomorphism

\[ K^\times/(K^{\times p} (1 + \pi^2\mathcal{O}_v)) \to \text{Hom}\Bigl(\frac{1}{\pi}\mathcal{O}_v/\wp(\mathcal{O}_v), \text{Br}(K)[p]\Bigr). \]

Suppose Kv has infinite dimension $\kappa$ as an $\mathbb{F}_p$ -vector space. Then $({1}/{\pi})\mathcal{O}_v/\wp(\mathcal{O}_v)$ also has dimension at least $\kappa$ , since it has $({1}/{\pi})\mathcal{O}_v/\mathcal{O}_v \cong Kv$ as a quotient. Consequently $\text{Hom}(({1}/{\pi})\mathcal{O}_v/\wp(\mathcal{O}_v), \text{Br}(K)[p])$ has cardinality at least $2^\kappa$ , as we can construct homomorphisms by freely choosing images of some fixed basis vectors.

On the other hand, $K^\times/(K^{\times p} (1 + \pi^2 \mathcal{O}_v))$ has cardinality at most $\kappa$ . Indeed, in the subgroup series

\[ K^\times \ge K^{\times p} \mathcal{O}_v^\times \geq K^{\times p} (1 + \pi \mathcal{O}_v) \geq K^{\times p} (1 + \pi^2 \mathcal{O}_v), \]

the first factor group $K^\times/(K^{\times p} \mathcal{O}_v^\times) \cong vK/p$ is countable, the second factor group

\[ (K^{\times p} \mathcal{O}_v^\times) / (K^{\times p} (1 + \pi \mathcal{O}_v)) \cong \mathcal{O}_v^\times / (\mathcal{O}_v^{\times p} (1 + \pi \mathcal{O}_v)) \cong (Kv)^\times/p, \]

has cardinality at most $\kappa$ , and the third factor group $(K^{\times p} (1 + \pi \mathcal{O}_v)) / (K^{\times p} (1 + \pi^2 \mathcal{O}_v))$ has cardinality at most $\kappa$ since it has a surjection from $(1 + \pi \mathcal{O}_v)/(1 + \pi^2 \mathcal{O}_v) \cong Kv$ . This shows that $K^\times/(K^{\times p} (1 + \pi^2 \mathcal{O}_v))$ has cardinality at most $\kappa$ . Since we cannot have a surjective map from a set of cardinality at most $\kappa$ to one of cardinality at least $2^\kappa$ , this is a contradiction.

Thus the residue field Kv is perfect. If the value group vK is p-divisible, every element of $K^\times$ can be written as the product of a pth power with an element of $1 + \mathfrak{m}_v$ . Lemma 4.1 implies that then $\left[{\mathcal{O}_v}, {K^\times}\right) = 0$ . By the pairing axiom (Pair), this is only possible if $\mathcal{O}_v \subseteq \wp(K)$ , implying that $Kv = \wp(Kv)$ , but this is in contradiction to the assumption that Kv is not p-closed. Therefore the value group vK is not p-divisible. Let w be the coarsening of v corresponding to the largest p-divisible convex subgroup of vK.

Since Kv has a $\mathbb{Z}/p$ -extension by assumption, there exists a $\mathbb{Z}/p$ -extension of K which is unramified with respect to v, and in particular unramified with respect to w, so Kw is not p-closed. Repeating the previous argument with (K,w), we see that the residue field Kw is perfect. By Lemma 4.3 (K,w) has a uniformiser $\pi$ , and by Lemma 4.4 its residue field Kw is finite. In particular, v cannot be a proper refinement of w, so $w = v$ .

The subgroup of vK generated by the value of the uniformiser $\pi$ is isomorphic to $\mathbb{Z}$ and convex. Consider the coarsening v’ of v with value group $v'K = vK/\langle v(\pi)\rangle$ (i.e. the finest proper coarsening of v). The residue field Kv’ carries a valuation with value group $\mathbb{Z}$ , and so Kv’ is imperfect. As above, Kv’ is also not p-closed. If v’ is not the trivial valuation, then Lemma 4.2 and Lemma 4.4 imply that Kv’ is finite, which is absurd. Therefore v’ must be trivial, and vK is isomorphic to $\mathbb{Z}$ .

By the construction of the pairing, this diagram is commutative. The map $K/\wp(K) \to \widehat{K}/\wp(\widehat{K})$ is surjective since K is dense in $\widehat{K}$ and $\wp(\widehat{K})$ is open, as $\wp(\widehat{K}) \supseteq \widehat{\mathfrak{m}}_v$ by Hensel’s Lemma. It is also injective since $\wp(\widehat{K}) = \wp(K) + \widehat{\mathfrak{m}}_v$ , and $\widehat{\mathfrak{m}}_v \cap K = \mathfrak{m}_v \subseteq \wp(K)$ by the p-henselianity assumption. The map $\text{Br}(K)[p] \to \text{Br}(\widehat{K})[p]$ is an isomorphism since both sides are isomorphic to $\mathbb{Z}/p$ and the image of $\left[{\mathcal{O}_v}, {\pi}\right)$ in $\text{Br}(\widehat{K})[p]$ is non-zero by standard facts on the Brauer group of local fields (cf. [Reference Gille and SzamuelyGS17, Remark 6.3.6]).

Using the pairing axiom (Pair) for both K and $\widehat{K}$ (where it holds by Proposition 2.2), we obtain that $K^\times/p \to \widehat{K}^\times/p$ is an isomorphism.

\[ y_0^p + \pi y_1^p + \dotsb + \pi^{p-1} y_{p-1}^p = z^p + \pi z^p x^p .\]

Since $\pi$ is also a p-basis of the field $\widehat{K}$ , we can compare coefficients to obtain $z = y_0 \in K$ and $y_1 = zx$ , so in particular $x = y_1/y_0 \in K$ , as desired.

Consider the local field $\mathbb{F}_q(\!(t)\!)$ and enumerate its elements as $(x_\alpha)_{\alpha < 2^{\aleph_0}}$ . Let $s \in \mathbb{F}_q(\!(t)\!)$ be an element such that s and t are algebraically independent. We construct an increasing chain of subfields $K_\beta$ of $\mathbb{F}_q(\!(t)\!)$ indexed by $\beta < 2^{\aleph_0}$ , where $K_0 = \mathbb{F}_q(t)$ , every $K_\beta$ has cardinality strictly less than $2^{\aleph_0}$ , $K_\beta$ is relatively algebraically closed in $\mathbb{F}_q(\!(t)\!)$ , the image of the map $K_\beta^\times/p \to \mathbb{F}_q(\!(t)\!)^\times/p$ contains the pth power classes of all $x_\alpha$ with $\alpha < \beta$ , and s is transcendental over $K_\beta$ . This can be done straightforwardly by transfinite induction: at limit ordinals $\beta$ one takes $K_\beta := \bigcup_{\gamma < \beta} K_\gamma$ , and for a successor ordinal $\beta+1$ one takes $K_{\beta+1}$ to be the relative algebraic closure of $K_\beta(x)$ for a suitably chosen element x in the pth power class $x_\beta \mathbb{F}_q(\!(t)\!)^{\times p}$ of $x_\beta$ . The only constraint is ensuring that s remains transcendental over $K_{\beta+1}$ ; to do so, it suffices to pick x not algebraic over $K_\beta(s)$ , which can always be done since $x_\beta \mathbb{F}_q(\!(t)\!)^{\times p}$ has strictly larger cardinality than $K_\beta(s)$ .

Now the field $K = \bigcup_{\beta < 2^{\aleph_0}} K_\beta$ is relatively algebraically closed in $\mathbb{F}_q(\!(t)\!)$ , the map $K^\times/p \to \mathbb{F}_q(\!(t)\!)^\times/p$ is an isomorphism (surjectivity is by construction, and injectivity follows from relative algebraic closedness), but K does not contain s. The restriction map of absolute Galois groups $\text{G}_{\mathbb{F}_q(\!(t)\!)} \to \text{G}_K$ is an isomorphism: here surjectivity follows from relative algebraic closedness, and injectivity holds since the restriction map $\text{G}_{\mathbb{F}_q(\!(t)\!)} \to \text{G}_{\mathbb{F}_q(t)}$ is injective by Krasner’s Lemma [Reference EfratEfr06, Corollary 18.5.3]. Furthermore, the restriction map of Brauer groups $\text{Br}(K) \to \text{Br}(\mathbb{F}_q(\!(t)\!))$ is an isomorphism, as $\text{Br}(K)$ is known from the general theory of the Brauer group [Reference GrothendieckGro68, Corollaire 2.3].

Hence K is a counterexample to Proposition 4.7 when the condition $[K : K^p] \leq p$ is dropped, and even to Theorem 1.1 without the axiom (Imp). Indeed, K is not isomorphic to a local field, since it carries a unique non-trivial henselian valuation (namely the restriction of the t-adic valuation on $\mathbb{F}_q(\!(t)\!)$ ) but is not complete.

We can now prove the main theorem from the introduction.

5. Sharpenings using quotients of the absolute Galois group

Let us observe the ways in which the axiom $\text{G}_K \cong \text{G}_{\mathbb{F}_q(\!(t)\!)}$ was used in the proof of Theorem 1.1. We established that K has characteristic p in Lemma 2.3, since the p-Sylow subgroups of $\text{G}_K$ are projective, and used Corollary 3.3 to obtain a valuation on K. Throughout § 4 we relied on K having countably many $\mathbb{Z}/p$ -extensions, i.e. $\text{Hom}(\text{G}_K, \mathbb{Z}/p)$ being countable. Lastly, we finished the proof by observing that non-isomorphic local fields of characteristic p have non-isomorphic absolute Galois groups.

The last two of these four uses of the axiom actually need much weaker information than the isomorphism of full absolute Galois groups $\text{G}_K \cong \text{G}_{\mathbb{F}_q(\!(t)\!)}$ . Indeed, the information about $\mathbb{Z}/p$ -extensions is contained in $\text{G}_K^{\mathrm{ab}}$ , and likewise non-isomorphic local fields of characteristic p have non-isomorphic abelianised absolute Galois groups by [Reference Neukirch, Schmidt and WingbergNSW08, Proposition 7.5.9].

In this section we want to show that Lemma 2.3 and Corollary 3.3 also have versions in which the hypothesis only refers to certain quotients of $\text{G}_K$ . The main tool to achieve this is [Reference Chebolu, Efrat and MináčCEM12]. The proofs here make slightly more serious use of Galois cohomology and Milnor K-groups than the preceding sections, so we assume a passing familiarity with the definitions and basic properties on the part of the reader.

Following [Reference Chebolu, Efrat and MináčCEM12, § 4], given a profinite group G and a prime p, we write $G^{[3,p]}$ for the quotient $G/((G^p[G,G])^p [G^p[G,G], G])$ , i.e. the quotient of G by the third term in its lower p-central series (also called its descending p-central sequence). Observe that $G^{[3,p]}$ is a pro-p group which is nilpotent of class 2 as a central extension of $G/(G^p[G,G])$ by $G^p[G,G]/((G^p[G,G])^p [G^p[G,G], G])$ .

Conversely, every field K of characteristic p satisfies the condition on $\text{G}_K$ .

Proof. Suppose first that K is of characteristic p. Let $H \leq \text{G}_K$ be a closed subgroup and $L/K$ the corresponding algebraic extension. Then the maximal pro-p quotient $F := H(p) = \text{G}_L(p)$ is free pro-p [Reference Neukirch, Schmidt and WingbergNSW08, Corollary 6.1.3]. Since the quotient $H^{[3,p]}$ of H is pro-p, the epimorphism $H \to F$ induces an isomorphism $H^{[3,p]} \cong F^{[3,p]}$ . This proves that $\text{G}_K$ satisfies the given condition.

Suppose conversely that K is a field of characteristic not p such that the condition on $\text{G}_K$ is satisfied; we shall show that $\text{Br}(K)[p] = 0$ . Let $L = K(\zeta_p)$ , and $H = \text{G}_L \leq \text{G}_K$ . The group $G_K/H \cong \text{Gal}(L/K)$ embeds into $(\mathbb{Z}/p)^\times$ and hence is cyclic of order dividing $p-1$ . Let F be a free pro-p group satisfying $H^{[3,p]} \cong F^{[3,p]}$ . By [Reference Fried and JardenFJ08, Corollary 23.1.2] there exists a field L’ containing $\mathbb{Q}(\zeta_p)$ with absolute Galois group $\text{G}_{L'}$ isomorphic to F. Now [Reference Chebolu, Efrat and MináčCEM12, Theorem A] implies that $H^2(\text{G}_L, \mathbb{Z}/p) \cong H^2(\text{G}_{L'}, \mathbb{Z}/p)$ , and the latter group vanishes since $\text{G}_{L'} \cong F$ is projective. Therefore $H^2(\text{G}_L, \mu_p) \cong H^2(\text{G}_L, \mathbb{Z}/p)$ also vanishes. Since the restriction map $H^2(\text{G}_K, \mu_p) \to H^2(\text{G}_L, \mu_p)$ is injective as $L/K$ is of degree coprime to p, $\text{Br}(K)[p] = H^2(\text{G}_K, \mu_p)$ must be trivial.

Let $\mathfrak{C}$ be the class of finite groups which are extensions of an abelian group by a nilpotent group of nilpotence class at most 2. In other words, $\mathfrak{C}$ consists of finite groups G for which there exists a normal subgroup H such that $G/H$ is abelian and the quotient of H by its centre is abelian. An arbitrary profinite group G always has a maximal pro- $\mathfrak{C}$ quotient $G(\mathfrak{C})$ in the usual sense [Reference Fried and JardenFJ08, Definition 17.3.2].

We now give a sharpened version of Corollary 3.3, concerning the existence of a valuation with residue field which is not p-closed. Compared to the earlier corollary, the Galois-theoretic assumption is weakened. On the other hand, we also do not obtain full henselianity, but only p-henselianity, which was a crucial property in § 4.

Our proof loosely follows [Reference Efrat and FesenkoEF99, Proposition 2.3], although somewhat more work is required here. We will use the notion of l-henselianity for a valuation v on a field K, where l is an arbitrary prime number [Reference Engler and PrestelEP05, § 4.2]. So far, we have only considered this for l the characteristic of K. In general, a valuation v on K is l-henselian if it extends uniquely to every Galois extension of l-power degree, or equivalently to every $\mathbb{Z}/l$ -extension [Reference Engler and PrestelEP05, Theorem 4.2.2]. This holds if and only if Hensel’s Lemma holds for every polynomial whose roots generate a Galois extension of l-power degree [Reference Engler and PrestelEP05, Theorem 4.2.3]. If the residue field Kv is of characteristic not equal to l and K contains the lth roots of unity, this is equivalent to $1 + \mathfrak{m}_v$ being contained in the set of lth powers [Reference Engler and PrestelEP05, Corollary 4.2.4]. This is an analogue of the previously used fact that, when l is the characteristic of K, v is l-henselian if and only if $\mathfrak{m}_v$ is contained in the image of the polynomial $X^l-X$ [Reference Chatzidakis and PereraCP17].

The following can be seen as a criterion for p-henselianity. It is inspired by the criterion for full henselianity [Reference EfratEfr95, Proposition 2.1].

Let $R \subseteq K'(\zeta_l)$ be the intersection of the valuation rings $\mathcal{O}_{w_i'}$ . By [Reference Engler and PrestelEP05, Theorem 3.2.7], the natural ring homomorphism $R \to \prod_{i=1}^p K'(\zeta_l)w_i'$ is surjective, so in particular $R^\times/l$ surjects onto the product of the $(K'(\zeta_l)w_i')^\times/l$ . Since the fields $K'(\zeta_l)w_i'$ are not l-closed, Kummer theory implies that the groups $(K'(\zeta_l)w_i')^\times/l$ are not trivial. Therefore $R^\times/l$ has size at least $l^p$ . On the other hand $K'(\zeta_l)^\times/R^\times$ surjects onto $K'(\zeta_l)^\times/\mathcal{O}_{w_1'}^\times$ , which is precisely the value group of $w_1'$ . This value group contains the value group of v as a subgroup of finite index, and therefore is not l-divisible since the value group of v is not l-divisible (see for instance [Reference EfratEfr06, Lemma 1.1.4(d)]). The short exact sequence $1 \to R^\times \to K'(\zeta_l)^\times \to K'(\zeta_l)^\times/R^\times \to 1$ induces an exact sequence $1 \to R^\times/l \to K'(\zeta_l)^\times/l \to (K'(\zeta_l)^\times/R^\times)/l \to 1$ since $K'(\zeta_l)^\times/R^\times$ is torsion-free. We deduce that $K'(\zeta_l)^\times/l$ has size at least $l^{p+1}$ , and so the field $K'(\zeta_l)$ has a $(\mathbb{Z}/l)^{p+1}$ -extension by Kummer theory. The extension $K'(\zeta_l)/K$ is abelian as the composite of the two abelian extensions $K'/K$ and $K(\zeta_l)/K$ .

We also require the following simple lemma for the proof of Proposition 5.3.

\[ \frac{p^{p^{n+1}}-1}{p^{p^n}-1} = \sum_{i=0}^{p-1} \big(p^{p^n}\big)^i \]

is congruent to p modulo $p^{p^n}-1$ , and therefore certainly coprime to $p^{p^n}-1$ . It follows that there are infinitely many primes l dividing $p^{p^n}-1$ for some n. Pick such a prime l for which $l \nmid q-1$ , $l \nmid [\mathbb{F}_q : \mathbb{F}_p]$ . We have $\zeta_l \not\in \mathbb{F}_q$ since the order $\lvert \mathbb{F}_q^\times\rvert = q-1$ is not a multiple of l. Let $m \geq 1$ be maximal such that $l^m \mid p^{p^n}-1$ , so that the cyclic group $\mathbb{F}_{p^{p^n}}^\times$ has elements of order $l^m$ , but none of order $l^{m+1}$ . This means that $\zeta_l, \zeta_{l^m} \in \mathbb{F}_{p^{p^n}}$ , but $\zeta_{l^{m+1}} \not\in \mathbb{F}_{p^{p^n}}$ . The degree $[\mathbb{F}_p(\zeta_l) : \mathbb{F}_p] | [\mathbb{F}_{p^{p^n}} : \mathbb{F}_p] = p^n$ is a power of p.

The degree $[\mathbb{F}_p(\zeta_{l^{m+1}}) : \mathbb{F}_p(\zeta_l)]$ is a power of l since the extension arises as an iterated Kummer extension by successively adjoining lth roots of the element $\zeta_l$ . The degree is also not 1 since $\mathbb{F}_p(\zeta_l) \subseteq \mathbb{F}_{p^{p^n}} \not\ni \zeta_{l^{m+1}}$ . Lastly, $[\mathbb{F}_p(\zeta_{l^{m+1}}) : \mathbb{F}_p(\zeta_l)] = [\mathbb{F}_q(\zeta_{l^{m+1}}) : \mathbb{F}_q(\zeta_l)]$ since $\mathbb{F}_q(\zeta_l)$ and $\mathbb{F}_p(\zeta_{l^{m+1}})$ are Galois extensions of $\mathbb{F}_p(\zeta_l)$ of coprime degrees.

Proof of Proposition 5.3. Choose l and m as in Lemma 5.5. Fix an isomorphism $\sigma \colon \text{G}_K(\mathfrak{C}) \to \text{G}_{\mathbb{F}_q(\!(t)\!)}(\mathfrak{C})$ . Let $L_0/K$ be the cyclic extension corresponding to $\mathbb{F}_q(\!(t)\!)(\zeta_l)/\mathbb{F}_q(\!(t)\!)$ via $\sigma$ , and let $L = L_0(\zeta_l)$ . Both $L_0/K$ and $K(\zeta_l)/K$ are abelian extensions of p-power degree by the choice of l, and hence so is their compositum $L/K$ . Let $F/\mathbb{F}_q(\!(t)\!)$ be the extension corresponding to L. Since $L/K$ and $F/\mathbb{F}_q(\!(t)\!)$ are abelian extensions, the Galois groups $\text{G}_L$ and $\text{G}_F$ have isomorphic maximal 2-step nilpotent quotients by the construction of $\mathfrak{C}$ . In particular, the graded rings $H^\ast(\text{G}_L, \mathbb{Z}/l) \cong \operatorname{K}^M_\ast(L)/l$ and $H^\ast(\text{G}_F, \mathbb{Z}/l) \cong \operatorname{K}^M_\ast(F)/l$ are isomorphic by [Reference Chebolu, Efrat and MináčCEM12, Theorem A]. Thus $L^\times/l = \operatorname{K}^M_1(L)/l \cong F^\times/l \cong (\mathbb{Z}/l)^2$ and $\operatorname{K}^M_2(L)/l \cong H^2(\text{G}_F, \mathbb{Z}/l) = \text{Br}(F)[l] \cong \mathbb{Z}/l$ . Since the ring $\operatorname{K}^M_\ast(L)/l$ is skew-commutative, the multiplication map $L^\times/l \times L^\times/l \to \operatorname{K}^M_2(L)/l$ is (surjective and) skew-commutative, so $\operatorname{K}^M_2(L)/l$ is naturally identified with the second exterior power $\bigwedge^2_{\mathbb{F}_l} L^\times/l$ . Thus the product of any $\mathbb{F}_l$ -linearly independent elements of $L^\times/l$ in $\operatorname{K}^M_2(L)/l$ is non-zero. Applying [Reference EfratEfr99, Main Theorem]Footnote ¹ with the prime l and $T = L^{\times l}$ , it follows that L carries a valuation v such that $(vL : l \cdot vL) \geq l$ , the residue field Lv has characteristic not l, and $1 + \mathfrak{m}_v \subseteq L^{\times l}$ , so (L,v) is l-henselian.

We claim that the residue field Lv does not contain $\zeta_{l^{m+1}}$ . If it does, then by l-henselianity $\zeta_{l^{m+1}} \in L$ , since $\zeta_{l^{m+1}}$ has l-power degree over $L \supseteq \mathbb{F}_p(\zeta_l)$ . We know that $L^\times/l \cong (\mathbb{Z}/l)^2$ , say generated by the classes of $a, b \in L^\times$ . Then by Kummer theory both $L(a^{1/l^{m+1}})$ and $L(b^{1/l^{m+1}})$ are $\mathbb{Z}/l^{m+1}$ -extensions of L. Their intersection is L, since their minimal non-trivial subextensions $L(a^{1/l})$ and $L(b^{1/l})$ are distinct. Therefore L has a $(\mathbb{Z}/l^{m+1})^2$ -extension, namely $L(a^{1/l^{m+1}}, b^{1/l^{m+1}})$ , and accordingly F also has a $(\mathbb{Z}/l^{m+1})^2$ -extension. By [Reference Neukirch, Schmidt and WingbergNSW08, Proposition 7.5.9], this implies that the residue field of F contains $\zeta_{l^{m+1}}$ . Since $F/\mathbb{F}_q(\!(t)\!)$ is a Galois extension of p-power degree, its residue field is a p-power degree extension of $\mathbb{F}_q$ . However, $\zeta_{l^{m+1}}$ has degree divisible by l over $\mathbb{F}_q$ by the choice of l. This contradiction shows that $\zeta_{l^{m+1}} \not\in Lv$ . The degree of $\zeta_{l^{m+1}}$ over Lv is some power of l, since $\zeta_{l^{m+1}}$ has l-power degree over $\mathbb{F}_p(\zeta_l) \subseteq Lv$ .

We now investigate the restriction of v to K. Observe that $\zeta_{l^{m+1}}$ generates an abelian extension of Kv whose degree is divisible by l since $\zeta_{l^{m+1}}$ has degree divisible by l even over Lv, and therefore Kv is not l-closed. If $v|_K$ is not p-henselian, then by Lemma 5.4 some finite abelian extension $K'/K$ has a $(\mathbb{Z}/l)^{p+1}$ -extension. However, the extension of $\mathbb{F}_q(\!(t)\!)$ corresponding to it under $\sigma$ (as indeed any finite extension of $\mathbb{F}_q(\!(t)\!)$ ) does not have a $(\mathbb{Z}/l)^{p+1}$ -extension [Reference Neukirch, Schmidt and WingbergNSW08, Proposition 7.5.9]. This contradiction shows the p-henselianity of $v|_K$ .

Lastly, let us show that $\zeta_l \not\in Kv$ . Suppose for a contradiction that $\zeta_l \in Kv$ , then also $\zeta_l \in K$ as $\zeta_l$ has p-power degree over $\mathbb{F}_p$ and $v|_K$ is p-henselian. Therefore K has a $(\mathbb{Z}/l)^2$ -extension, given by on the one hand adjoining an lth root of an element $x \in K^\times$ with $v(x) \in vK$ not divisible by l, and on the other hand adjoining an lth power root of unity (since we know that $\zeta_{l^{m+1}}$ does not lie in Lv and therefore not in K). However, $\mathbb{F}_q(\!(t)\!)$ does not have a $(\mathbb{Z}/l)^2$ -extension, so given the isomorphism $\sigma$ we once more obtain a contradiction. Therefore $Kv(\zeta_l)/Kv$ is a non-trivial Galois extension which is of p-power degree by the choice of l. Thus Kv is not p-closed.

We can now deduce the following strengthened version of Theorem 1.1.

• (Gal₀) the maximal pro- $\mathfrak{C}$ quotients $\text{G}_K(\mathfrak{C})$ and $\text{G}_{\mathbb{F}_q(\!(t)\!)}(\mathfrak{C})$ are isomorphic, where $\mathfrak{C}$ is (as above) the class of finite groups which are extensions of an abelian group by a nilpotent group of nilpotence class at most 2;

• (Imp) K has exponent of imperfection at most 1;

• (Brau) $\text{Br}(K)[p] \cong \mathbb{Z}/p\mathbb{Z}$ ; and

• (Pair) the natural pairing $\text{Hom}(\text{G}_K, \mathbb{Z}/p) \times K^\times/p \to \text{Br}(K)[p]$ induces an isomorphism $K^\times/p \cong \text{Hom}(\text{Hom}(\text{G}_K, \mathbb{Z}/p), \text{Br}(K)[p])$ .

Then K is isomorphic to the local field $\mathbb{F}_q(\!(t)\!)$ .

The choice of the class of groups $\mathfrak{C}$ is not quite optimal. It is already apparent from the proofs above that it could be shrunk by some amount, strengthening the statement of Theorem 5.7. However, this would yield a class less easy to describe than $\mathfrak{C}$ .