## Biquadratic extensions with one break.(English)Zbl 1033.11054

Let $$K$$ be a finite extension of the field of $$2$$-adic numbers, and consider totally ramified biquadratic extensions $$N/K$$ with Galois group $$G$$. The ramification filtration of $$G$$ (with lower numbering) contains one or two breaks; the Galois module structure of ideals in extensions with two breaks were studied by the second author [Can. J. Math. 50, 1007–1047 (1998; Zbl 1015.11056)]. In this article, the authors treat the more complicated case of fields with one break and show explicitly how the ideals $${\mathfrak P}^i$$ in $$N$$ decompose into indecomposable $${\mathbb Z}_2[G]$$-modules.

### MSC:

 11S15 Ramification and extension theory 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 20C11 $$p$$-adic representations of finite groups

### Keywords:

Galois module structure; local fields; ramification groups

Zbl 1015.11056
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