mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Abhyankar's lemma (named after

Shreeram Shankar Abhyankar Shreeram Shankar Abhyankar (22 July 1930 – 2 November 2012) was an Indian American mathematician known for his contributions to algebraic geometry. He, at the time of his death, held the Marshall Distinguished Professor of Mathematics Chair ...

) 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 field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...

s such that ''A'' and ''B'' are

finite extension In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory &mdash ...

s of ''C'', with ramification indices ''a'' and ''b'', and ''B'' is tamely ramified over ''C'' and ''b'' divides ''a'', then the

compositum In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subf ...

''AB'' is an unramified extension of ''A''.

References

*. Theorem 3, page 504. *. *
p. 279
* . Theorems in algebraic geometry Lemmas in algebra Algebraic number theory Theorems in abstract algebra {{algebra-stub

See also

References