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- 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 = qb + r and either r = 0 or v(r) < v(b).
The function v is 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.
The name comes from the fact that the extended Euclidean algorithm can be carried out in any Euclidean domain.
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