Discrete Valuation Ring - Examples

Examples

Let Z(2)={ p/q : p, q in Z, q odd }. Then the field of fractions of Z(2) is Q. Now, for any nonzero element r of Q, we can apply unique factorization to the numerator and denominator of r to write r as 2kp/q, where p, q, and k are integers with p and q odd. In this case, we define ν(r)=k. Then Z(2) is the discrete valuation ring corresponding to ν. The maximal ideal of Z(2) is the principal ideal generated by 2, and the "unique" irreducible element (up to units) is 2.

Note that Z(2) is the localization of the Dedekind domain Z at the prime ideal generated by 2. Any localization of a Dedekind domain at a non-zero prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define rings Z(p) for any prime p in complete analogy.

For an example more geometrical in nature, take the ring R = { f/g : f, g polynomials in R and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined in a neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is X and the valuation assigns to each function f the order (possibly 0) of the zero of f at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line.

Another important example of a DVR is the ring of formal power series R = K] in one variable T over some field K. The "unique" irreducible element is T, the maximal ideal of R is the principal ideal generated by T, and the valuation ν assigns to each power series the index of the first non-zero coefficient.

If we restrict ourselves to real or complex coefficients, we can consider the ring of power series in one variable that converge in a neighborhood of 0 (with the neighborhood depending on the power series). This is also a discrete valuation ring.

Finally, the ring Zp of p-adic integers is a DVR, for any prime p. Here p is an irreducible element; the valuation assigns to each p-adic integer x the largest integer k such that pk divides x.

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