Abhyankar's inequality is an inequality involving extensions of valued fields in algebra, introduced by Abhyankar (1956).

Abhyankar's inequality states that for an extension K/k of valued fields, the transcendence degree of K/k is at least the transcendence degree of the residue field extension plus the rank of the quotient of the valuation groups; here the rank of an abelian group ${\displaystyle A}$ is defined as ${\displaystyle \dim _{\mathbb {Q} }(A\otimes \mathbb {Q} )}$.

References

• Abhyankar, Shreeram (1956), "On the valuations centered in a local domain", American Journal of Mathematics, 78 (2): 321–348, doi:10.2307/2372519, ISSN 0002-9327, JSTOR 2372519, MR 0082477