I) unit polynomials with non-vanishing degree
is its own multiplicative inverse, showing that does not hold in a general commutative Ring with one.
This phenomenon is uniquely characterized by the following equivalence:
Proof. Negated replication.
Let be a unit polynomial of non-vanishing degree . Let denote its multiplicative inverse, i.e. .
Claim. The polynomial has non-vanishing degree .
Proof. Suppose . Since , it follows from that is a zero divisor. However, at the same time implies that is a unit, arriving at a contradiction.
Since both , one concludes as well as .
Existence of the desired ring elements is assured by the following construction.
Let rise discretely.
If , implying , holds, since the construction arrived at this point, one finds
The above condition is met for at least one , since otherwise would imply , which is impossible since and is a unit element.
By construction, as well as are given.
Setting , one calculates