Graded and Valued Field Extensions

Abstract

We pursue in this chapter the investigation of valuations through the graded structures associated to the valuation filtration. The graded field of a valued field is an enhanced version of the residue field, inasmuch as it encapsulates information about the value group in addition to the residue field. It thus captures much of the structure of the field, particularly in the Henselian case. This point is made clear in §5.2, where we show that—when the ramification is tame—Galois groups and their inertia subgroups of Galois extensions of valued fields can be determined from the corresponding extension of graded fields. Henselian fields are shown to satisfy a tame lifting property from graded field extensions, generalizing the inertial lifting property. In §5.1, we lay the groundwork for the subsequent developments by an independent study of graded fields, their algebraic extensions and their Galois theory.

Appendices

Exercises

Exercise 5.1

(A graded field not of group-ring type) Let k be an arbitrary field and let F be a graded field with F ₀=k and $\Gamma_{\mathsf {F}}=\mathbb {Z}[\frac{1}{2}]$. For $i\in \mathbb {N}$, let y _i ∈F be a nonzero homogeneous element of degree 2⁻ⁱ, and let ${u_{i}=y_{i+1}^{2}y_{i}^{-1}\in k^{\times}}$. Consider the sequence

$$(u_0, u_0u_1^2, u_0u_1^2u_2^4,\ldots, \operatorname*{ \mathchoice{\textstyle\prod}{\prod}{\prod}{\prod}}_{i=0}^n u_i^{2^i}, \ldots) \in \operatorname*{ \mathchoice{\textstyle\prod}{\prod}{\prod}{\prod}}_{i=0}^\infty \bigl(k^\times/(k^\times)^{2^{i+1}} \bigr). $$

This sequence defines an element u _∞ in the projective limit $\varprojlim\bigl(k^{\times}/(k^{\times})^{2^{i+1}}\bigr)$ for the canonical maps $k^{\times}/(k^{\times})^{2^{i}}\leftarrow k^{\times}/(k^{\times})^{2^{j}}$ that arise from the inclusions $(k^{\times})^{2^{i}}\supseteq (k^{\times})^{2^{j}}$ for i≤j. Let $k^{\times}/k^{\times2^{\infty}}=\varprojlim \bigl(k^{\times}/(k^{\times})^{2^{i+1}}\bigr)$ and let ${\varphi\colon k^{\times}\to k^{\times}/k^{\times 2^{\infty}}}$ be the canonical map.

• (a) Show that modulo the image of φ the element u _∞ does not depend on the choice of the homogeneous elements y _i , and that F is of group-ring type if and only if u _∞ lies in the image of φ.

• (b) (Hwang–Wadsworth) Let $k=\mathbb {Q}$, fix a prime number p, and let ${u_{i}=p^{2^{i}}}$ for $i\in \mathbb {N}$. Show that the corresponding element $u_{\infty}\in \mathbb {Q}^{\times}/\mathbb {Q}^{\times 2^{\infty}}$ satisfies $u_{\infty}^{3}=\varphi(p)^{-1}$. Use u _∞ to construct a graded field F with $\mathsf {F}_{0}=\mathbb {Q}$ and $\Gamma_{\mathsf {F}}=\mathbb {Z}[\frac{1}{2}]$ that is not of group-ring type.

Exercise 5.2

(An extension of the quotient field of a graded field that is not the quotient field of a graded field) (Van Geel–Van Oystaeyen) Let F ₀ be any field of characteristic different from 2, and let F be the Laurent polynomial ring F=F ₀[t,t ⁻¹], with the $\mathbb {Z}$-grading by the degree of t, so F _γ =F ₀ t ^γ for each $\gamma\in \mathbb {Z}$, and q(F)=F ₀(t). Let $E=q(\mathsf {F})(\sqrt{t+1})$. Show that there is no graded field extension K of F such that q(K)≅E. As a consequence, show that there is no algebraic graded field extension K of F such that q(K) contains an isomorphic copy of E. [Hint: if E↪q(K), show that one can assume K/F is tame, and use Prop. 5.35.]

Exercise 5.3

Let K be an algebraic graded field extension of a graded field  F.

• (i) Show that there is a one-to-one correspondence between the unramified extensions L of F in K and the field extensions E of F ₀ in K ₀, which maps L to L ₀ and E to $E\otimes_{\mathsf {F}_{0}}\mathsf {F}$.

• (ii) Show that if K/F is totally ramified, there is a one-to-one correspondence between the graded field extensions L of F in K and the subgroups Δ such that Γ _F ⊆Δ⊆Γ _K , which maps L to Γ _L and Δ to ⨁ _δ∈Δ K _δ .

• (iii) Give an example of an extension K/F containing totally ramified extensions L ₁, L ₂ of F with L ₁≠L ₂ and $\Gamma_{\mathsf {L}_{1}}=\Gamma_{\mathsf {L}_{2}}$. (Thus, the correspondence in (ii) does not generally hold if K/F is not totally ramified.)

Exercise 5.4

There are characterizations of normal graded field extensions analogous to characterizations for ungraded fields. Let F and K be graded fields with F⊆K⊆F _alg . Prove that the following conditions are equivalent:

• (a) K is normal over F;

• (b) for any homogeneous a∈K, the minimal polynomial of a over q(F) splits over K;

• (c) K is a graded splitting field over F, i.e., there is a family of homogenizable polynomials {f _i } _i∈I in F[X] such that each f _i splits over K and K  is generated over F by the roots of the f _i ;

• (d) for every F-homomorphism η:K→F _alg , we have η(K)⊆K.

Exercise 5.5

Let K be a normal graded field extension of a graded field  F. Let T be the tame closure of F in K, and let I be the purely inseparable closure of F in K, i.e., the graded subfield of K generated over F by all the homogeneous elements of K which are purely inseparable over F. Prove that T is Galois over F with $\operatorname {\mathcal {G}}(\mathsf {T}/\mathsf {F}) \cong \operatorname {\mathit{Aut}}(\mathsf {K}/\mathsf {F})$; I is the fixed graded field $\mathsf {K}^{\operatorname {\mathit{Aut}}(\mathsf {K}/\mathsf {F})}$; and K=T⊗ _F I.

Exercise 5.6

Let (K,w) be an extension of the valued field (F,v) with K  normal over F. Prove that $\operatorname {\mathsf {gr}}_{w}(K)$ is normal over $\operatorname {\mathsf {gr}}_{v}(F)$ and $\overline{K}$ is normal over $\overline{F}$.

Exercise 5.7

Let (F,v) be a valued field and f∈F[X] be a monic λ-uniform polynomial. Assume $\widetilde{f}^{(\lambda)}$ is irreducible, hence f is irreducible by Lemma 5.47, and let L=F[X]/(f). Let w be any valuation on L extending v. Show that

$$\operatorname {\mathsf {gr}}_w(L)\,\cong_g \,\operatorname {\mathsf {gr}}(F)[X]^{(\lambda)}/(\widetilde{f}^{(\lambda)}). $$

Deduce that w is the unique extension of v to L and that v is defectless in L.

Exercise 5.8

Let F be a graded field. A finite-dimensional commutative graded algebra S over F is said to be separable if S is a direct product of graded fields, each tame over F. Prove:

• (i) A finite-dimensional commutative graded algebra S is separable over F if and only if q(S) is separable over q(F) (i.e., a direct product of fields each separable over q(F)).

• (ii) Let S and S′ be finite-dimensional graded field extensions of F. Then, S⊗ _F S′ is separable over F if and only if S and S′ are each separable over F.

• (iii) Suppose S is a finite-dimensional commutative semisimple graded F-algebra and K is a graded field extension of F. Then S⊗ _F K is separable over K if and only if S is separable over F.

Exercise 5.9

Let F⊆K be fields with $[K{\mspace{1mu}:\mspace{1mu}}F] < \infty$ and K Galois over F. Let v ₁, …, v _r be the valuations on K extending a valuation v on F. Let $\alpha = \operatorname* {\mathit{min}}(v_{1}, \ldots, v_{r})$, which is a surmultiplicative v-value function on K, with ${\operatorname {\mathsf {gr}}_{\alpha}(K) \cong_{g} \operatorname {\mathsf {gr}}_{v_{1}}(K) \times \ldots \times \operatorname {\mathsf {gr}}_{v_{r}}(K)}$ by Th. 4.36. Let $G = \operatorname {\mathcal {G}}(K/F)$. Since G acts transitively on the v _i , all the graded fields $\operatorname {\mathsf {gr}}_{v_{i}}(K)$ are isomorphic. Moreover, for any ${\sigma \in \operatorname {\mathcal {G}}(K/F)}$, since σ permutes the v _i , we have α∘σ=α. Hence, σ induces a graded $\operatorname {\mathsf {gr}}(F)$-automorphism $\widetilde{\sigma}$ of $\operatorname {\mathsf {gr}}_{\alpha}(K)$ defined on homogeneous elements by ${\widetilde{\sigma}(\widetilde{c}) = \widetilde {\sigma(c)}}$ for all c∈K. Thus, there is a group homomorphism $\Xi\colon \operatorname {\mathcal {G}}(K/F) \to \operatorname {\mathit{Aut}}_{\operatorname {\mathsf {gr}}(F)}(\operatorname {\mathsf {gr}}_{\alpha}(K))$ given by $\sigma \mapsto \widetilde{\sigma}$.

• (i) Let $G = \operatorname {\mathcal {G}}(K/F)$, and let

  $${G^\mathit{ram}\,= \, \{\sigma\in G \mid v_1(\sigma(c) - c)) > v_1(c)\ \text{for all}\ c\in K^\times\}}, $$

  which is the ramification group for the extension v ₁ of v. Prove that

  $$\operatorname {\mathit{ker}}(\Xi) \,= \, \textstyle \bigcap\limits_{\sigma \in G} \sigma\, G^\mathit{ram}\,\sigma ^{-1}. $$

• (ii) Prove that the fixed ring $\operatorname {\mathsf {gr}}(K)^{\Xi(G)}$ is a graded field isomorphic to the purely inseparable closure of $\operatorname {\mathsf {gr}}(F)$ in $\operatorname {\mathsf {gr}}_{v_{i}}(K)$.

• (iii) Prove that α is a tame v-gauge (see Prop. 6.41 in the next chapter) if and only if Ξ is injective and $\operatorname {\mathsf {gr}}_{\alpha}(K)$ is Ξ(G)-Galois over $\operatorname {\mathsf {gr}}(F)$.

Exercise 5.10

Let (K,v _K ) be a valued field extension of a valued field  (F,v). Assume K is a Galois extension of F and v _K is the unique extension of v to K. Show that the following conditions are equivalent:

• (a) K is tamely ramified over F;

• (b) ${\operatorname {\mathcal {G}}^{ \mathit{ram}}(K/F) = \{\operatorname {\mathit{id}}\}}$;

• (c) K is defectless over F and $\operatorname {\mathsf {gr}}(K)$ is Galois over $\operatorname {\mathsf {gr}}(F)$;

• (d) The ρ of Prop. 5.51 defines an isomorphism $\operatorname {\mathcal {G}}(K/F) \!\stackrel{\sim}{\to}\! \operatorname {\mathcal {G}}(\operatorname {\mathsf {gr}}(K)/\!\operatorname {\mathsf {gr}}(F))$.

Exercise 5.11

Let F be a field with valuation v and let K be a cyclic Galois field extension of F such that v has a unique and tamely ramified extension to K. Let $e = \lvert \Gamma_{K}{\mspace{1mu}:\mspace{1mu}}\Gamma_{F}\rvert$. Prove that Γ _K /Γ _F is a cyclic group and that $\overline{F}$ contains a primitive e-th root of unity.

Notes

§5.1: This section is mostly an expanded version of Hwang–Wadsworth [102] and Mounirh–Wadsworth  [167, §1]. There was earlier work by Van Geel–Van Oystaeyen [250], who considered algebraic extensions of graded fields, but only with $\mathbb {Z}$-gradings. Another precursor is the paper by Boulagouaz [24]. He considered graded field extensions, and showed that the minimal polynomial of a homogeneous element in a graded field extension is homogenizable. For a graded field F, he also pointed out the valuation on q(F) determined by a total ordering on Γ _F . He proved Prop. 5.19 and Prop. 5.50(iii) in [24, Th. 4, Th. 5]. Subsequently, Boulagouaz  [28] investigated the Galois theory of graded field extensions and proved portions of Prop. 5.32 for finite-degree Galois extensions.

§5.2: Much of subsection 5.2.1 comes from Mounirh–Wadsworth [167, §1]. Preceding [167], Boulagouaz  [27] had considered uniform polynomials over Henselian fields and introduced $\widetilde{f}^{(\lambda)}$ for λ-uniform polynomials; he proved a version of Lemma 5.46 and (a) ⇒ (b) and (a) ⇒ (e) of Th. 5.49. Cor. 5.56 was proved by Hwang–Wadsworth [102, Th. 5.2], with a better proof in [167, Cor. 1.13].

Exercise 5.1 is based on Hwang–Wadsworth  [102, Ex. 1.2]. Exercise 5.2 is adapted from Van Geel–Van Oystaeyen  [250, Ex. 3.10.2]. Exercise 5.6 comes from Mounirh–Wadsworth  [167, Th. 1.5].