Vector spaces must be equipped with one of these, and Galois theory describes how their extension correspond to permutation groups.
Ax and Kochen name an isomorphism theorem between two of these objects. A valuation is defined over one of these structures, and these structures necessarily have a Krull dimension of zero. Adjoining a primitive root of unity to the rational numbers gives a cyclotomic one of these, while these are called "splitting" if a polynomial factors completely over them. All of these with finite order have order equal to the power of some prime, and are isomorphic to each other. Vector spaces must be equipped with one of these, and Galois theory describes how their extension correspond to permutation groups. FTP, name these mathematical structure equiped with addition, multiplication, and division.
Ax and Kochen name an isomorphism theorem between two of these objects.
{Fields} [accept "{p}-{adic numbers}" before "{valuation}"]