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Wiktionary英語版での「algebraically independent」の意味 |
algebraically independent
形容詞
algebraically 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,
- 2006, Alexander B. Levin, Difference algebra, 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.
使用する際の注意点
- Perhaps unexpectedly, a single element of may be said to be algebraically independent (over ).
反意語
- (which does not または whose elements do not satisfy any nontrivial polynomial equation over a given field): algebraically dependent
Further reading
- Algebraic independence on Wikipedia.
- Algebraic independence on Encyclopedia of Mathematics
- Algebraically Independent on Wolfram MathWorld
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