algebraic extension

plural algebraic extensions

1. field theoryalgebraA field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K).

   We describe the correspondence between subfields of the field P#95;a(O) that are algebraic extensions of the field K(X) and the subgroups of finite index of the fundamental group #92;pi#95;1(X#92;setminusO).

   2013, Askold Khovanskii, Galois Theory, Coverings, and Riemann Surfaces, Springer, page 65

   2 more examples

   What we now have to prove is that: if an overring R#42; of R is such that R#42; is an integral domain, R#42; is integral over R and the quotient field #92;mathfrak#123;L#125;#42; of R#42; is a finite algebraic extension of #92;mathfrak#123;L#125;, then R#42; is a local ring and a finite R-module.

   1964, Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, page 200

   It turns out that when an irreducible polynomial splits in an algebraic extension, then the order at the new irreducible factors remains the same as before for arguments that are defined over the ground field.

   2005, Manuel Bronstein, Symbolic Integration I: Transcendental Functions, 2nd edition, Springer, page 110