Let $A$ and $B$ be two polynomials in $\mathbf Z[X]$ which generate $\mathbf Z[X]$, that is assume that there exist polynomials $U$ and $V$ in $\mathbf Z[X]$ such that $$ A \cdot U + B \cdot V=1. $$ ...
This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves. ...