Let 
be a field, and 
a 
-algebra. Elements 
, ..., 
are algebraically independent over 
if the natural surjection ![K[Y_1,...,Y_n]->K[y_1,...,y_n]](/images/equations/AlgebraicallyIndependent/Inline7.svg)
is an isomorphism. In other
words, there are no polynomial relations 
with coefficients in 
.
Algebraically Independent
This entry contributed by Johnny Chen
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References
Reid, M. Undergraduate Commutative Algebra. Cambridge, England: Cambridge University Press, 1995.SeeAlso
Irrational Number, Lindemann-Weierstrass Theorem, Schanuel's Conjecture, Shidlovskii Theorem, Transcendental Number
Referenced on Wolfram|Alpha
Algebraically IndependentCite this as:
Chen, Johnny. "Algebraically Independent." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AlgebraicallyIndependent.html