algebraically independent en · ADJ
Meanings
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(not-comparable, qualifier:field theory) (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.
1999, David Mumford, The Red Book of Varieties and Schemes: Includes the Michigan Lectures, Springer, Lecture Notes in Mathematics 1358, 2nd Edition, Expanded, page 40, If the statement is false, there are n elements x_1,…,x_n in R such that their images ◌̅x_i in R/P are algebraically independent. Let 0 ne p∈P. Then p,x_1,…,x_n cannot be algebraically independent over k, so there is a polynomial P(Y,X_,…,X_n) over k such that P(p,x_,…,x_n)=0.
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.
Setting y#95;i#61;y#95;#123;i,1#125; (where 1 denotes the identity of the semigroup T) we obtain a #92;sigma-algebraically independent over R set #92;#123;y#95;i#92;verti#92;inI#92;#125; such that S#61;R#92;#123;(y#95;i)#95;#123;i#92;inI#125;#92;#125;.
2006, Alexander B. Levin, “Difference algebra”, in M. Hazewinkel, editor, Handbook of Algebra, Volume 4, Elsevier (North-Holland), page 251: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.
If α ne 0,1 is algebraic and β is an algebraic irrational of degree d>2, then αᵝ,…,α are algebraically independent.
2014, M. Ram Murty, Purusottam Rath, Transcendental Numbers, Springer, page 138, Let us begin with the following conjecture of Schneider
Antonyms
algebraically dependent (antonym(s) of “which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field”)
Translations (1)
it algebricamente indipendente (which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field)