Euclidean Domain - Examples

Examples

Examples of Euclidean domains include:

• Any field. Define f(x) = 1 for all nonzero x.
• Z, the ring of integers. Define f(n) = |n|, the absolute value of n.
• Z, the ring of Gaussian integers. Define f(a + bi) = a2 + b2, the squared norm of the Gaussian integer a + bi.
• Z (where ω is a cube root of 1), the ring of Eisenstein integers. Define f(a + bω) = a2 − ab + b2, the norm of the Eisenstein integer a + bω.
• K, the ring of polynomials over a field K. For each nonzero polynomial P, define f(P) to be the degree of P.
• K], the ring of formal power series over the field K. For each nonzero power series P, define f(P) as the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f(P)≤f(Q) iff P divides Q.
• Any discrete valuation ring. Define f(x) to be the highest power of the maximal ideal M containing x (equivalently, to the power of the generator of the maximal ideal that x is associated to). The previous case K] is a special case of this.
• A Dedekind domain with finitely many nonzero prime ideals P₁, ..., P_n. Define, where is the discrete valuation corresponding to the ideal P_i. (Samuel 1971)