algebraically independent

Definitions

1. 1

   (Of a subset S of the extension field L of a given field extension L / K) whose elements…

   (Of a subset S of the extension field L of a given field extension L / K) whose elements do not satisfy any non-trivial polynomial equation with coefficients in K.

   • The singleton set #92;#123;#92;alpha#92;#125; is algebraically independent over K if and only if the element #92;alpha is transcendental over K.
   • A subset S#92;subsetL is algebraically independent over K if every element of S is transcendental over K and over each of the extension fields over K generated by the remaining elements of S.

The neighborhood

• antonymalgebraically dependentantonym(s) of “which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field”