Let R be a ring of all real numbers , Show that H= { m+nV2 | m, ň E Z} is a subring of R.

ring theory

11

Let f=z" + an-12¹-1 + ... + a₁ € C[z] be a polynomial of degree n ≥ 1. Which of the factorisations below must exist? Select one or more: a. f=(z-C₁)(z-C₂)(z- Cn), where C₁, ..., Cn are complex numbers b. f= (z-C₁)(z-C₂)(z-Cn), where C₁, . ...., Cn are different complex numbers c. f=g₁(z) gk(z), where g₁, . ...., 9k € C[z] each have exactly one root, and all these roots are different d. f = (z² + r₁Z+S₁)... (z² + rkz + Sk), where r₁, ..., rk and S₁, ..., Sk are real numbers e. f=g₁(z) gk(z), where g₁, , ..., 9k € C[z] each have exactly one root, and their degrees are all different