Definition 1

Let E be an extension of a field K; a family $\mathbf{x} = (x_i)_{i \in I}$ of elements of E is said to be algebraically free over K if the monomials $\mathbf{x}^a = \prod_{i \in I} x_i^{a_i}$ with respect to the $x_i$ (for $a = (\alpha_i)_{i \in I}$, in $\mathbf{N}^{(I)}$) are linearly independent over K. In the contrary case the family is said to be algebraically related over K.

Definition 1 may also be expressed as follows: