Determinants

Consider $${{x}_{0}},{{x}_{1}}, \cdots ,{{x}_{n}} \in {{K}^{n}}$$ If $${{x}_{1}}, \cdots ,{{x}_{n}}$$ are independent,we have i $${{x}_{0}} = {{x}_{1}}{{t}_{1}} + \cdots + {{x}_{n}}{{t}_{n}}$$ for unique t1i…, t n These are rational functions of the elements, as can be inferred from their determination by the elimination process, homogeneous linear in xo. Transposition between x0 and xi replaces t iby 1/t i and t j (j≠i) by -t j/t i Hence there exists a polynomial b, homogeneous linear and antisymmetric in n vector arguments, unique up to a constant multiplier, such that $${{t}_{i}} = \delta \left( {{{x}_{1}}, \ldots ,{{x}_{0}} \ldots {{,}_{1}}{{x}_{n}}} \right)/\delta \left( {{{x}_{1}}, \ldots {{x}_{i}}, \ldots ,{{x}_{n}}} \right).$$ The constant multiplier can be chosen to give it the value $$\delta ({{l}_{1}}, \ldots ,{{l}_{n}}) = 1,$$ in respect to the elements of the fundamental base in K n .