Abhyankar's lemma

Short description: Allows one to kill tame ramification by taking an extension of a base field

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.

See also

• Finite extensions of local fields

References

• Cornell, Gary (1982), "On the Construction of Relative Genus Fields", Transactions of the American Mathematical Society 271 (2): 501–511, doi:10.2307/1998895 . 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 , 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 .