algebraically independent
English
editAdjective
editalgebraically independent (not comparable)
- (algebra, 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.
- The singleton set is algebraically independent over if and only if the element is transcendental over .
- A subset is algebraically independent over if every element of is transcendental over and over each of the extension fields over generated by the remaining elements of .
- 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 elements in such that their images in are algebraically independent. Let . Then cannot be algebraically independent over , so there is a polynomial over such that .
- 2006, Alexander B. Levin, “Difference algebra”, in M. Hazewinkel, editor, Handbook of Algebra, Volume 4, Elsevier (North-Holland), page 251:
- Setting (where 1 denotes the identity of the semigroup ) we obtain a -algebraically independent over set such that .
- 2014, M. Ram Murty, Purusottam Rath, Transcendental Numbers, Springer, page 138,
- Let us begin with the following conjecture of Schneider:
- If is algebraic and is an algebraic irrational of degree , then
- are algebraically independent.
Usage notes
edit- Perhaps unexpectedly, a single element of may be said to be algebraically independent (over ).
Antonyms
edit- (antonym(s) of “which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field”): algebraically dependent
Translations
editwhich does not or whose elements do not satisfy any nontrivial polynomial equation over a given field
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