algebraically independent

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English

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Adjective

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algebraically independent (not comparable)

  1. (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

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  • Perhaps unexpectedly, a single element of may be said to be algebraically independent (over ).

Antonyms

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  • (antonym(s) of which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field): algebraically dependent

Translations

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Further reading

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