  Many a concrete theorem of abstract algebra admits a short and elegant proof
by contradiction but with Zorn's Lemma (ZL). A few of these theorems have
recently turned out to follow in a direct and elementary way from the Principle
of Open Induction distinguished by Raoult. The ideal objects characteristic of
any invocation of ZL are eliminated, and it is made possible to pass from
classical to intuitionistic logic. If the theorem has finite input data, then a
finite partial order carries the required instance of induction, which thus is
constructively provable. A typical example is the well-known theorem "every
nonconstant coefficient of an invertible polynomial is nilpotent".
