algebraically independent

English

Adjective

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 $\{\alpha \}$ is algebraically independent over $K$ if and only if the element $\alpha$ is transcendental over $K$.

   A subset $S\subset L$ 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 $S$.

   • 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 $x_{1},\dots ,x_{n}$ in $R$ such that their images ${\overline {x}}_{i}$ in $R/P$ are algebraically independent. Let $0\neq p\in P$. 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 $P(p,x_{,}\dots ,x_{n})=0$.

   • 2006, Alexander B. Levin, “Difference algebra”, in M. Hazewinkel, editor, Handbook of Algebra, Volume 4, Elsevier (North-Holland), page 251:

     Setting $y_{i}=y_{i,1}$ (where 1 denotes the identity of the semigroup $T$) we obtain a $\sigma$-algebraically independent over $R$ set $\{y_{i}\vert i\in I\}$ such that $S=R\{(y_{i})_{i\in I}\}$.

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

     Let us begin with the following conjecture of Schneider:

     If $\alpha \neq 0,1$ is algebraic and $\beta$ is an algebraic irrational of degree $d\geq 2$, 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

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

Translations

which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field

• Italian: algebricamente indipendente m or f

Further reading

• Algebraic independence on Wikipedia.Wikipedia
• Algebraic independence on Encyclopedia of Mathematics
• Algebraically Independent on Wolfram MathWorld