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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.

References

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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