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In mathematics, Abhyankar's lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field.

More precisely, Abhyankar's lemma states that if A, B, C are local fields such that A and B are finite extensions of C, with ramification indices a and b, and B is tamely ramified over C and b divides a, then the compositum AB is an unramified extension of A.


Cornell, Gary (1982), "On the Construction of Relative Genus Fields", Transactions of the American Mathematical Society 271 (2): 501–511, JSTOR 1998895. Theorem 3, page 504.
Gold, Robert; Madan, M. L. (1978), "Some applications of Abhyankar's lemma", Mathematische Nachrichten 82: 115–119, doi:10.1002/mana.19780820112.
Grothendieck, A. (1971), Revêtements étales et groupe fondamental (SGA 1, Séminaire de Géométrie Algébriques du Bois-Marie 1960/61), Lecture Notes in Mathematics 224, Springer-Verlag, arXiv:math.AG/0206203, p. 279.
Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3rd ed.), Berlin: Springer-Verlag, p. 229, ISBN 3-540-21902-1, Zbl 1159.11039.

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