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Let $N/K$ be a Galois extension with Galois group $𝒢$. We study the set $𝒯\left(𝒢\right)$ of $\mathbb{Z}$-linear combinations of characters in the Burnside ring $ℬ\left(𝒢\right)$ which give rise to $\mathbb{Z}$-linear combinations of trace forms of subextensions of $N/K$ which are trivial in the Witt ring W$\left(K\right)$ of $K$. In particular, we prove that the torsion subgroup of $ℬ\left(𝒢\right)/𝒯\left(𝒢\right)$ coincides with the kernel of the total signature homomorphism.

We give several examples of classes of trace forms for which the ideal of annihilating polynomials is principal. We prove, that in general, the annihilating ideal is not a principal ideal.

Let ${\mathbb{Z}}_d\wr$m$bethegeneralizedalternatinggroup.Weprovethatalldoublecoversof$ℤd ≀ m$occurasGaloisgroupsoveranyalgebraicnumberfield.WefurtherrealizesomeofthesedoublecoversastheGaloisgroupsofregularextensionsof\mathbb{Q}\left(T\right).Ifdisoddandm>7,theneverycentralextensionof$ℤd ≀ m$occursastheGaloisgroupofaregularextensionof\mathbb{Q}\left(T\right).Wefurtherimprovesomeofourearlierresultsconcerningdoublecoversofthegeneralizedsymmetricgroup$ℤd ≀ m$.$