Let d and n be positive integers and let K be a totally real number field of discriminant d and degree n. We construct a theta series \(\theta _{K} \in {\mathcal {M}}_{d,n}\), where \({\mathcal {M}}_{d,n}\) is a space of modular forms defined in terms of n and d. Moreover, if d is square free and n is at most 4 then \(\theta _{K}\) is a complete invariant for K. We also investigate whether or not the collection of \(\theta \)-series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant d, is a linearly independent subset of \({\mathcal {M}}_{d,4}\). This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well.

令d和n为正整数，令K为判别d和度n的全实数字段。我们在{\ mathcal {M}} _ {d，n} \）中构造一个theta系列\（\ theta _ {K} \，其中\（{\ mathcal {M}} _ {d，n} \）是由n和d定义的模块化形式的空间。此外，如果d为无平方且n最多为4，则\（\ theta _ {K} \）是K的完全不变式。我们还将调查\（\ theta \）的集合与固定平方自由判别式d的四次数域的同构类集相关联的-系列是\（{\ mathcal {M}} _ {d，4} \）的线性独立子集。如果数域的次数小于或等于3，这就是事实。我们提供了计算和启发式的证据，表明在4级时，这些theta系列也应该是独立的。