:As I recall, in order for a subring of a Euclidean domain to be an ideal, it would absolutely have to contain 0; the " I " you describe above necessarily does "'not "'contain 0, hence cannot be an ideal.
42.
Since the " norm " function is not defined for the zero element of the ring, we consider the degree of the polynomial " f " ( " x " ) = 0 to also be undefined so that it follows the rules of a norm in a euclidean domain.
43.
An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but when an explicit algorithm for Euclidean division is known, one may use Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and B�zout's identity.
44.
This follows from the fact that \ mathbb { C } [ Z ] ( the polynomial ring over the complex numbers in the indeterminant " Z " ) is a Euclidean domain; a well-known truth about polynomial rings over fields ( in short, you can " perform the Euclidean algorithm on the set of polynomials with complex coefficients ).
45.
This proof builds on Lagrange's result that if p = 4n + 1 is a prime number, then there must be an integer " m " such that m ^ 2 + 1 is divisible by " p " ( we can also see this by Euler's criterion ); it also uses the fact that the Gaussian integers are a unique factorization domain ( because they are a Euclidean domain ).
46.
So, given an integral domain " R ", it is often very useful to know that " R " has a Euclidean function : in particular, this implies that " R " is a PID . However, if there is no " obvious " Euclidean function, then determining whether " R " is a PID is generally a much easier problem than determining whether it is a Euclidean domain.
47.
If " R " is an integral domain that is not a field then " R " [ " X " ] is neither a Euclidean domain nor a principal ideal domain; however it could still be a unique factorization domain ( and will be so if and only it " R " itself is a unique factorization domain, for instance if it is "'Z "'or another polynomial ring ).
