# AAED2

From Example Problems

Let be a Euclidean Domain with a function . Prove that

(a)

For all , the property of the function gives

Therefore, has the minimum value.

(b)

Let .

(): 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
so .

Therefore, .

(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 .