Geodesic
Galois module structure of square power classes for biquadratic extensions
Canadian journal of mathematics, Tome 75 (2023) no. 3, pp. 804-827

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DOI

For a Galois extension $K/F$ with $\text {char}(K)\neq 2$ and $\mathrm {Gal}(K/F) \simeq \mathbb {Z}/2\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}$, we determine the $\mathbb {F}_{2}[\mathrm {Gal}(K/F)]$-module structure of $K^{\times }/K^{\times 2}$. Although there are an infinite number of (pairwise nonisomorphic) indecomposable $\mathbb {F}_{2}[\mathbb {Z}/2\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}]$-modules, our decomposition includes at most nine 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.
DOI : 10.4153/S0008414X22000165
Mots-clés : Biquadratic extension, Galois module, Hilbert 90, pro-p groups, absolute Galois groups, Klein 4-group 
Chemotti, Frank; Mináč, Ján; Schultz, Andrew; Swallow, John. Galois module structure of square power classes for biquadratic extensions. Canadian journal of mathematics, Tome 75 (2023) no. 3, pp. 804-827. doi: 10.4153/S0008414X22000165
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     title = {Galois module structure of square power classes for biquadratic extensions},
     journal = {Canadian journal of mathematics},
     pages = {804--827},
     year = {2023},
     volume = {75},
     number = {3},
     doi = {10.4153/S0008414X22000165},
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