We determine the shapes of all degree $4$ number fields that are Galois. These lie in four infinite families depending on the Galois group and the tame versus wild ramification of the field. In the $V_4$ case, each family is a two-dimensional space of orthorhombic lattices and we show that the shapes are equidistributed, in a regularized sense, in these spaces as the discriminant goes to infinity (with respect to natural measures). We also show that the shape is a complete invariant in some natural families of $V_4$-quartic fields. For $C_4$-quartic fields, each family is a one-dimensional space of tetragonal lattices and the shapes make up a discrete subset of points in these spaces. We prove asymptotics for the number of fields with a given shape in this case.

中文翻译：

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伽罗瓦四次场的形状

我们确定伽罗瓦度数为 $4$ 的所有数字字段的形状。根据伽罗瓦群和该领域的驯服与狂野分支，它们位于四个无限家族中。在 $V_4$ 的情况下，每个族都是正交晶格的二维空间，我们证明了在这些空间中，随着判别式趋于无穷大（相对于自然度量），形状在正则化意义上是等分布的。我们还表明，在 $V_4$-四次域的一些自然族中，形状是完全不变量。对于 $C_4$-四次域，每个族是四方格子的一维空间，形状构成这些空间中点的离散子集。在这种情况下，我们证明了具有给定形状的场数的渐近性。