Algebraic independence means the absence of a nonzero polynomial relation over the base field. It extends the distinction between algebraic and transcendental elements to finite tuples.
Elements of a field extension of are algebraically independent over when no nonzero polynomial in vanishes at .
The rational function field is the field of fractions of when the variables are algebraically independent over .
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Algebraic independence is a concept from algebraic geometry and number theory that describes a certain property of numbers, functions, or algebraic entities. It refers to a set of elements that cannot satisfy any non-trivial polynomial relations with rational (or integer) coefficients.