An elementary proof of Wiebe's Theorem

This note has three contributions. The first one is to give elementary proofs about finite free resolution using only the notion of latent regular element (regular element which may be obtained after addition of indeterminates) and no notion from homological algebra. The second contribution is to use this to provide a simple proof of Wiebe's Theorem, a fundamental result in the theory of resultants [5]. Finally the third contribution is to explain how to “automatically” prove results or obtain counter-examples about regular sequences of a given length. We obtain in this way a new example of a local ring with a regular sequence b1,b2 such that b2,b1 is not regular.