Rost Motives and H90

This chapter introduces the notion of a Rost motive, which is a summand of the motive of a Rost variety 𝑋. It highlights the theorem that, assuming that Rost motives exist and H90(n − 1) holds, then 𝐻𝑛+1 ét(𝑘, ℤ(𝑛)) injects into 𝐻𝑛+1 ét(𝑘(𝑋), ℤ(𝑛)). While there may be many Rost varieties associated to a given symbol, there is essentially only one Rost motive. The Rost motive captures the part of the cohomology of a Rost variety 𝑋. Since a Rost motive is a special kind of symmetric Chow motive, the chapter begins by recalling what this means. It then introduces the notion of 𝔛-duality. This duality plays an important role in the axioms defining Rost motives, as well as a role in the construction of the Rost motive in the next chapter. Finally, this chapter assumes that Rost motives exist and proves a key theorem.

The N-Th Canonical Roots of Matrices

After a deep investigation on the nth canonical roots of matrices, we establish the key theorem concerning how to find the n-th canonical roots. Relationship between canonical roots of a complex matrix and its n-th roots is also investigated.. Existence of canonical roots is not true for arbitrary matrix. By investigating the extension rule, this paper introduces the concept of canonical roots and gives an algorithm determining the form of canonical roots.