Let be a Euclidean Domain with a function . Prove that
For all , the property of the function gives
Therefore, has the minimum value.
(): Let . Then, . So, by property of the function
Since is the minimum value, . Therefore,
, and so .
(): Let . Then ,
and so has the minimum value. Now, since is a Euclidean Domain, there exists such that , with or . Since has the minimum value, is impossible, and so .
Thus, , and is a unit with inverse . Therefore, , and
(c) Use (b) to determine and
For , with the absolute value, , and so .
For , with , for
all . Thus .
For , with , , and so non-zero polynomials of degree zero .
For , with , , and so .