Uniformizers

In abstract algebra, a uniformizer is a concept primarily used in the study of local fields and valuation theory. Specifically, within the context of a discrete valuation ring (DVR), a uniformizer is a special element that generates the unique maximal ideal of the ring. A discrete valuation ring is a commutative ring with a single maximal ideal, and this ideal is principal, meaning it can be generated by a single element. This generator is called a uniformizer.

Let R be a discrete valuation ring and m be its unique maximal ideal. Then m = (π) for some element π ∈ R. This element π is called a uniformizer of R. The term "uniformizer" arises because this element plays a crucial role in defining a valuation on the field of fractions of R. The valuation assigns to each non-zero element of the field a non-negative integer representing the highest power of the uniformizer that divides the element. This valuation provides a notion of "size" or "order of divisibility" in a way that is consistent with the ring structure.

Uniformizers are fundamental to understanding the structure of local fields. For instance, in the ring of p-adic integers, denoted Z_p, the prime number p itself serves as a uniformizer. The maximal ideal of Z_p is (p), and every non-zero integer in Z_p can be uniquely written as a unit multiplied by a power of p. This structure is deeply connected to the p-adic valuation. The existence and properties of uniformizers are key characteristics that distinguish discrete valuation rings from other types of rings.