Algebraically independent

English

Adjective

1. Template:Lb (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.

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   • 1999, Template:W, The Red Book of Varieties and Schemes: Includes the Michigan Lectures, Springer, Template:W 1358, 2nd Edition, Expanded, page 40,

     If the statement is false, there are $n$ elements $x_1,\dots ,x_n$ in $R$ such that their images ${\bar{x}}_i$ in $R/P$ are algebraically independent. Let $0\ne p\in P$. Then $p,x_1,\dots x_n$ cannot be algebraically independent over $k$, so there is a polynomial $P(Y,X_{,}\dots ,X_n)$ over $k$ such that $P(p,x_{,}\dots ,x_n)=0$.

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   • 2014, M. Ram Murty, Purusottam Rath, Transcendental Numbers, Springer, page 138,

     Let us begin with the following conjecture of Schneider:

     If $\alpha \ne 0,1$ is algebraic and $\beta$ is an algebraic irrational of degree $d\ge 2$, then

     ${\alpha }^{\beta },\dots ,{\alpha }^{{\beta }^{d-1}}$

     are algebraically independent.

Usage notes

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

Antonyms

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Translations

• Italian: Template:T

Further reading

• Template:Pedia
• Algebraic independence on Template:W
• Algebraically Independent on Template:W