Galois module structure of square power classes for biquadratic extensions

Abstract

For a Galois extension $K/F$ with $\text{char}(K)\ne 2$ and $\text{Gal}(K/F)\simeq \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$, we determine the ${\mathbb{F}}_2[\text{Gal}(K/F)]$-module structure of $K^{\times }/K^{\times 2}$. Although there are an infinite number of (pairwise non-isomorphic) indecomposable ${\mathbb{F}}_2[\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}]$-modules, our decomposition includes at most $9$ indecomposable types. This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representation theory allows for an infinite number of indecomposable types.

arXiv commentv1: 26 pages. v2: 23 pages. Theorem 1 includes an additional summand type; corresponding adjustments appear in section 3. Section 6 contains new results on realizability. Miscellaneous typos corrected

References in corpus (8)

• Galois module structure of Milnor K-theory in characteristic p
• Galois module structure of Galois cohomology for embeddable cyclic extensions of degree p^n
• Galois module structure of square power classes for biquadratic extensions
• Galois module structure of pth-power classes of cyclic extensions of degree p^n
• Galois Module Structure of $p$th-Power Classes of Extensions of Degree p
• Galois Module Structure of \Z/\ell^n-th Classes of Fields
• Galois module structure of Milnor K-theory mod p^s in characteristic p
• Parameterizing solutions to any Galois embedding problem over Z/p^nZ with elementary p-abelian kernel

Cited by in corpus (3)

• Galois module structure of square power classes for biquadratic extensions
• On the indecomposability of a remarkable new family of modules appearing in Galois theory
• Arithmetic properties encoded in the Galois module structure of $K^{\times }/K^{\times p^m}$