Discrete valuation ring

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Dissecting a Discrete Valuation Ring (DVR)

What it is
- A DVR is a ring that is a domain, a principal ideal domain (every ideal is generated by one element), and local with exactly one nonzero maximal ideal.
- There is a special element π in the ring, called a uniformizer, such that the maximal ideal is (π). Every nonzero element can be written uniquely as a unit times a power of π.

Key idea
- For any nonzero x in the ring, you can write x = u · π^k, where u is a unit (invertible) and k ≥ 0. The number k is the valuation v(x). The ring’s valuation captures how many times π divides x.
- The residue field is κ = R/(π), and the fraction field is Frac(R).

Examples
- Z localized at 2, denoted Z_(2): a DVR where the uniformizer is 2 and v measures the power of 2 in a rational number.
- p-adic integers Z_p for any prime p: π = p.
- The ring of formal power series k[[T]] in one variable over a field k: π = T, and v(f) is the position of the first nonzero coefficient.
- Related examples include other complete DVRs like the real/complex power series rings near 0 (with appropriate convergence).

Why it’s useful
- DVRs arise naturally by focusing on local behavior: localizing a Dedekind domain at a prime gives a DVR.
- They provide a simple way to study local properties of curves and functions, using the uniformizer and the valuation.

More structure you might encounter
- Fraction field: Frac(R)
- Residue field: κ = R/(π)
- Topology: DVRs carry a natural, often useful, topology; complete DVRs (like Z_p or k[[T]]) are especially important in number theory and algebraic geometry.

In short
- A DVR is a local PID with a single maximal ideal generated by a uniformizer. Every nonzero element factors uniquely as a unit times a power of this uniformizer, and the exponent is the valuation.