Suppose that Z, n ≥ 1 is a martingale with respect to a family of increasing o-fields F, n 2 1. Let G₁, be another family of increasing o-fields such that 9, CF, Vn 2 1. Suppose that Z,, is also G-measurable. Prove that Z₁ is a martingale with respect to G₁, n 21.

The field Z_2(u) is a splitting field for f(x)=x^3+x+1 over Z_2. This polynomial is separable. If u is a root, then find the two other roots of f(x) as powers of u

Exercise 6. Find primitive elenants in (a) the field Z,, (8) the ficeld Z13.

Suppose A € M₂(F) has multiplicative order 5. First, show that x³ – 1 = (x² − 4x + 1)(x² + 5x + 1) in F19 [x], where F19 is the field with 19 elements, i.e. Z19. Use this fact to determine all similarity classes of elements of M₂(F19) of multiplicative order 5.

Please answer #10

Lemma 60: If a ∈ Z then a · 1 = a. Exercise 28: Prove Lemma 60.