In
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
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
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 compa ...
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 — ...
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 su ...
''AB'' is an unramified extension of ''A''.
See also
*
Finite extensions of local fields In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups.
In th ...
References
*. Theorem 3, page 504.
*.
*
p. 279
* .
Theorems in algebraic geometry
Lemmas in algebra
Algebraic number theory
Theorems in abstract algebra
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