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Some self-dual local rings of integers not free over their associated orders. (English) Zbl 0737.11037

Let \(p\) be a prime number and let \(K\) be a finite extension of the field \(\mathbb Q_ p\) of \(p\)-adic numbers. Let also \(L/K\) be a finite abelian extension with Galois group \(G\) and let \({\mathcal O}_ L,{\mathcal O}_ K\) denote the valuation rings of \(L\) and \(K\) respectively. The associated order \({\mathcal A}_{L/K}\) is defined by: \({\mathcal A}_{L/K}=\{\alpha\in KG\mid \alpha{\mathcal O}_ L\subset{\mathcal O}_ L\}\). In their book [Elliptic functions and rings of integers. Prog. Math. 66. Boston et al.: Birkhäuser) (1987; Zbl 0608.12013)], Ph. Cassou-Noguès and M. Taylor ask if \({\mathcal O}_ L\) is free as an \({\mathcal A}_{L/K}\)-module when \({\mathcal O}_ L\) is isomorphic as an \({\mathcal O}_ K\) \(G\)-module to the inverse different \({\mathcal D}^{-1}_{L/K}\) or, equivalently, when \({\mathcal O}_ L\) is self-dual as an \({\mathcal O}_ K\) \(G\)-lattice.
The purpose of this paper is to construct, for each prime number \(p\) and for suitable extensions \(K/\mathbb Q_ p\), an elementary abelian extension \(L/K\) of degree \(p^ 2\) which yields a negative answer to this question. The extension \(L/K\) is totally ramified and \(K/\mathbb Q_ p\) is ramified. This last condition is necessary, in view of an unpublished result of D. Burns quoted by the author, which says that the question of Cassou- Noguès and Taylor has an affirmative answer if \(K/\mathbb Q_ p\) is unramified.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11S15 Ramification and extension theory
11S23 Integral representations
11S20 Galois theory
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References:

[1] Washington, Introduction to Cyclotomic Fields (1982) · Zbl 0484.12001 · doi:10.1007/978-1-4684-0133-2
[2] Serre, Local Fields (1979) · doi:10.1007/978-1-4757-5673-9
[3] Berg?, Ann. Inst. Fourier (Grenoble) 28 pp 17– (1978) · Zbl 0377.12009 · doi:10.5802/aif.715
[4] Ferton, C. R. Acad. Sci. Paris 276 pp A1483– (1973)
[5] Cassou-Nogu?s, Elliptic Functions and Rings of Integers 66 (1987)
[6] Martel, C. R. Acad. Sci. Paris 278 pp A117– (1974)
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