Meaning of algebraic extension | Babel Free

Definitions

A 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).

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.”

“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.”

“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).”

CEFR level