## Field of iterated Laurent series and its Brauer group.(English)Zbl 1452.16020

Let $$F$$ be an infinite field, $$\mathrm{Br}(F)$$ its Brauer group, $$p$$ a prime number, $$_p\mathrm{Br}(F) = \{\beta \in\mathrm{Br}(F): p\beta = 0\}$$, and $$\mathrm{Brd}_p(F)$$ the Brauer $$p$$-dimension of $$F$$, i.e., the supremum $$d \le \infty$$ of those integers $$n \ge 0$$, for which there exists a central division $$F$$-algebra $$\Delta$$ of $$p$$-primary exponent exp$$(\Delta)$$ and degree deg$$(\Delta) =\exp(\Delta)^n$$. It is well known that if $$F$$ contains a primitive $$p$$-th root of unity $$\rho$$ and $$A$$ is a cyclic $$F$$-algebra of degree $$p$$, then $$A$$ can be presented as $$F[x, y: x^p = \alpha, y^p = \beta, yx = \rho x, y]$$, for some $$\alpha, \beta \in F^{\ast}$$; we denote this presentation by $$(\alpha, \beta)_{p,F}$$. When $$\mathrm{char}(F) = p$$, every cyclic $$F$$-algebra of degree $$p$$ takes the form $$[\alpha, \beta)_{p,F} = F \langle x, y: x^- x = \alpha, y^p = \beta, yxy^{-1} = x + 1 \rangle$$, for some $$\alpha \in F$$, $$\beta \in F^{\ast}$$. These forms are called (Hilbert) symbol presentations of the algebras, and the algebras are also called symbol algebras. It is known that $$_p\mathrm{Br}(F)$$ is generated by the Brauer equivalence classes of cyclic $$F$$-algebras of degree $$p$$ in the following two cases: if $$F$$ contains a primitive $$p$$-th root of unity (see [A. S. Merkur’ev and A. A. Suslin, Math. USSR, Izv. 21, 307–340 (1983; Zbl 0525.18008); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011–1046 (1982)]); if $$\mathrm{char}(F) = p$$ (see Ch. VII, Theorem 28, in: [A. A. Albert, Structure of algebras. American Mathematical Society (AMS), Providence, RI (1939; JFM 65.0094.02)]). The symbol length $$\mathrm{Sym}(b_p)$$ of an element of $$_p\mathrm{Br}(F)$$ is the minimal number of symbol algebras needed to express it, and the symbol length of $$_p\mathrm{Br}(F)$$ is the supremum $$\mathrm{Sym}_p(F)$$ of $$\mathrm{Sym}(b_p): b_p \in{}_{p}\mathrm{Br}(F)$$.
The main results of the paper under review are stated and proved in its Sections 3 and 4. They concern the special case where $$F = k_n := k((\alpha_1)) \dots ((\alpha_n))$$ is the iterated Laurent formal power series field in $$n$$ variables over a field $$k$$. Theorem 3.5, the main result of Section 3, states that if $$k$$ is a perfect field with $$\mathrm{char}(k) = p$$, $$k_{sep}$$ is a separable closure of $$k$$, and $$m$$ is the rank (as a pro-$$p$$-group) of the Galois group $$\mathcal{G}(k(p)/k)$$ of the maximal $$p$$-extension $$k(p)$$ of $$k$$ in $$k_{\mathrm{sep}}$$, then $$\mathrm{Sym}_p(F) = n - 1$$, provided that $$m < n$$, and $$\mathrm{Sym}_p(F) = n$$ if $$m \ge n$$. As noted by the author, this complements the known result (presented by Proposition 3.1 of the paper) that $$\mathrm{Sym}_{p'}(F) = [n/2]$$ in case $$k$$ is algebraically closed, $$p'$$ is prime,and $$p' \neq\mathrm{char}(k)$$. When $$n = 3$$, these results indicate that $$\mathrm{Brd}_2(F) = 1$$ if and only if $$\mathrm{char}(k) \neq 2$$. This fact is generalized in Section 4 as follows: (i) If $$F = k_{n+1}$$, where $$k$$ is an algebraically closed field with $$\mathrm{char}(k) \neq 2$$, then $$I^n F$$ is linked, i.e., every two anisotropic bilinear $$n$$-fold Pfister forms over $$F$$ share an $$(n-1)$$-fold Pfister factor; (ii) when $$k$$ is a field of characteristic $$2$$ and $$F = k_{n+1}$$, $$I_q^n F$$ is not linked, i.e., there exists a pair of quadratic $$n$$-fold Pfister forms which do not share an $$(n - 1)$$-fold Pfister factor; (iii) for any field $$F$$ with $$\mathrm{char}(F) = 2$$ and degree $$[F: F^2] > 2^n$$, $$I^n F$$ is not linked, i.e., there exists a pair of anisotropic bilinear Pfister forms over $$F$$, which do not share an $$(n - 1)$$-fold factor.
Reviewer’s remark. Let $$(F, v)$$ be a Henselian valued field with a residue field $$\widehat F$$, and let $$\mathrm{Br}(\widehat F)_{p}$$ be the $$p$$-component of $$\mathrm{Br}(\widehat F)$$. Then $$\mathrm{Brd}_p(F) = \mathrm{Sym}_p(F)$$ in the following two cases: (i) if $$\mathrm{Br}(\widehat F)_p = \{0\}$$ and $$\widehat F$$ contains a primitive $$p$$-th root of unity (see (4.7), Theorem 2.3, and Corollary 5.6 of the reviewer’s paper in [J. Pure Appl. Algebra 223, No. 1, 10–29 (2019; Zbl 1456.16015)]; (ii) if $$(F, v)$$ is maximally complete, $$[F: F^p] = p^n$$, for some $$n \in \mathbb{N}$$, and $$\widehat F$$ is perfect (see Proposition 3.5 in the reviewer’s paper in: [Serdica Math. J. 44, 303–328 (2018)]). This recovers the proof of Proposition 3.1 and generalizes Theorem 3.5 of the paper under review to the case where $$(F, v)$$ is maximally complete with $$\mathrm{char}(F) = p$$, $$\widehat F$$ perfect and $$[F: F^n] = p^n$$.

### MSC:

 16K20 Finite-dimensional division rings 16S35 Twisted and skew group rings, crossed products 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 11E81 Algebraic theory of quadratic forms; Witt groups and rings 11E04 Quadratic forms over general fields

### Citations:

Zbl 0525.18008; Zbl 1456.16015; JFM 65.0094.02
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### References:

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