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# Euclidean domain

In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used.

More precisely, a Euclidean domain is an integral domain D for which can be defined a function v mapping nonzero elements of D to non-negative integers and possessing the following properties:

• For all nonzero a and b in D, v(ab) ≥ v(a).
• If a and b are in D and b is nonzero, then there are q and r in D such that a = bq + r and either r = 0 or v(r) < v(b).

The function v is variously called a gauge, valuation or norm. Note that some authors define the function in an inequivalent way which nonetheless still gives the same class of rings.

Examples of Euclidean domains include:

• Z, the ring of integers. Define v(n) = |n|, the absolute value of n.
• Z[i], the ring of Gaussian integers. Define v(z) = |z|2.
• K[X], the ring of polynomials over a field K. For each nonzero polynomial f. Define v(f) to be the degree of f.
• K[[X]], the ring of formal power series over the field K. For each nonzero power series f, define v(f) as the degree of the smallest power of X occurring in f.
• Any field. Define v(x) = 1 for all nonzero x.

Every Euclidean domain is a principal ideal domain. In fact, if I is a nonzero ideal of a Euclidean domain D and a nonzero a in I is chosen to minimize g(a), then I = aD.

The name comes from the fact that the extended Euclidean algorithm can be carried out in any Euclidean domain.

All Wikipedia text is available under the terms of the GNU Free Documentation License

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