Wikipedia's definition:
"In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K."
My textbook's definition:
"Let A be a subring of the commutative ring B. We say that the
elements $b_1, . . . , b_n$ of B are algebraically independent over A if the evaluation map
$ε_{b_1,...,b_n} : A[X_1, . . . , X_n] → B$ that evaluates each $X_i$ at $b_i$ is injective."
I'm a bit confused about how these two definitions are related. I think wikipedia's definiton is clearer but I'm not sure how this relates to the definition in the textbook. In other words, why are we looking at this evaluation map in order to understand the definition of algebraic independence?
Thanks in advance
Answer
The evaluation map at a point $(b_1,\ldots,b_n)\in B^n$ takes a polynomial $f(x_1,\ldots,x_n)\in A[x_1,\ldots,x_n]$ to its value at the point: $f(b_1,\ldots,b_n)\in B$. Clearly the zero polynomial will map to zero, but if the $b_i$ are algebraically independent by the first definition, then only the zero polynomial will map to zero, making the evaluation map injective.
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