According to this web page around at 1925 E. Artin conjectured that the discriminant of a number field determines the number field uniquely (up to isomorphism). This is also claimed in this article and in this slides.
The thing is I find rather hard to believe that Artin could have ever conjectured such a thing, as counter-examples are not so hard to find (consider the fields $\mathbb{Q}(\sqrt[3] {12})$ and $\mathbb{Q}(\sqrt[3] {6})$). So is there any actual records of Artin making that conjecture?.
In his first article after his thesis on quadratic function fields, namely "On the zeta functions of certain algebraic number fields" (Math. Ann. 1923), Artin cites Dedekind's article on pure cubic number fields in which the examples you mention figure prominently.
When Hasse proved in 1930 that there are discriminants D for which there are arbitrarily many nonconjugate cubic fields with discriminant D he did not mention Artin's name. I couldn't find anything related to this "conjecture" in Hasse's correspondence or in his diaries.