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 Wi 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, ...).
Read more about this topic: Witt Vector
Famous quotes containing the words construction of, construction and/or rings:
“There is, I think, no point in the philosophy of progressive education which is sounder than its emphasis upon the importance of the participation of the learner in the formation of the purposes which direct his activities in the learning process, just as there is no defect in traditional education greater than its failure to secure the active cooperation of the pupil in construction of the purposes involved in his studying.”
—John Dewey (1859–1952)
“No real “vital” character in fiction is altogether a conscious construction of the author. On the contrary, it may be a sort of parasitic growth upon the author’s personality, developing by internal necessity as much as by external addition.”
—T.S. (Thomas Stearns)
“We will have rings and things, and fine array,
And kiss me, Kate, we will be married o’ Sunday.”
—William Shakespeare (1564–1616)