Witt Vector - Construction of Witt Rings

Construction of Witt Rings

Fix a prime number p. A Witt vector over a commutative ring R is a sequence (X₀, X₁,X₂,...) of elements of R. Define the Witt polynomials W_i by

and in general

Then Witt showed that there is a unique way to make the set of Witt vectors over any commutative ring R into a ring, called the ring of Witt vectors, such that

• the sum and product are given by polynomials with integral coefficients that do not depend on R, and
• Every Witt polynomial is a homomorphism from the ring of Witt vectors over R to R.

The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,

(X₀, X₁,...) + (Y₀, Y₁,...) = (X₀+Y₀, X₁ + Y₁ + (X₀p + Y₀p − (X₀ + Y₀)p)/p, ...)

(X₀, X₁,...) × (Y₀, Y₁,...) = (X₀Y₀, X₀pY₁ + Y₀pX₁ + p X₁Y₁, ...).