We consider families of quartic number fields whose normal closures over $\mathbb{Q}$ have Galois group isomorphic to $D_4$, the symmetries of a square. To any such field $L$, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image $D_4$. We determine the asymptotic number of such $D_4$-quartic fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order-4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant.

Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with $L$-function methods. Both of these strategies fail in the case of $D_4$-quartic fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of $D_4$ combined with both approaches mentioned above to obtain exact asymptotics.

我们考虑在 $\mathbb{Q}$ 上正常闭包的四次数域族具有与 $D_4$ 同构的伽罗瓦群，即正方形的对称性。对于任何这样的字段 $L$，可以将相应二维不可约伽罗瓦表示的 Artin 导体与图像 $D_4$ 相关联。我们确定了这种由导体排序的 $D_4$-四次场的渐近数，并将主项明确计算为质量公式，验证了 Kedlaya 和 Wood 的启发式方法。此外，我们能够对任何有限数量的素数（有时，无限数量的素数）施加任何局部分裂条件，因此，我们还计算了类群和窄类中 4 阶元素的渐近数由判别式排序的二次域组。

传统上，有两种计算四次域的方法，一种是将算术不变理论与数几何技术相结合，另一种是将 Kummer 理论与 $L$ 函数方法相结合。这两种策略在由导体排序的 $D_4$-四次场的情况下都失败，因为计算包含具有大判别式的二次子场的四次场是困难的。然而，当按导体排序时，我们利用 $D_4$ 的外部自同构产生的额外代数结构结合上述两种方法来获得精确的渐近性。