Kummer first published in 1844 the property that the unique decomposition theorem in cyclotomic domain is not valid. 1847, the article was published in joseph liouville magazine. In the following years 1846 and 1847, Cuomo published his main theorem: the unique decomposition theorem of ideal prime numbers.

Kummer's concept of ideal number was independently developed by Kroneck and Dai Dejin in the following forty years. Dai Dejin encountered great difficulties when trying to directly popularize the concept of ideal number, which eventually led to his development of modular theory and idealism. Kroneck deepened type theory (the generalization of quadratic form) and factor theory to solve this problem. Dai Dejin's theory developed to later ring theory and abstract algebra, and Kroneck's theory became a powerful tool in algebraic geometry.