One of Kronecker's lesser dreams was to uniquely determine a polynomial from the knowledge of how many roots it has modulo every prime, where he obtained some partial results. One of the corollaries of Chebotarev's density theorem expands upon this idea and shows that a Galois extension of Q is characterized in full by the number of prime ideals it has of every norm, which is the final step in Neukirch's original proof of the Neukirch-Uchida theorem for Galois extensions.  For general number fields, however, a prime ideal count is insufficient to determine it, and some additional invariants come into play. We discuss many invariants related to number fields and determining capabilities, and extend these ideas to characterizing morphisms between number fields as well.

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