Find all generators and all subgroups of zx29 does it

Question

Exercise 1- Find all generators and all subgroups of Z^x₂₉. Does it contain a subgroup isomorphic to Z⁺₇? If yes, find some isomorphism.

Exercise 2- Is the set M = {a + b√5: a, b ∈ Q} with classical number addition and multiplication a field? Prove your answer. If it is a field, find another field to which it is isomorphic and give the isomorphism.

Exercise 3- Let f and g be two permutations over 9 elements, where

f = (2, 4, 5, 6, 3, 1, 8,9, 7) a            g = (8, 1, 5, 2, 6, 3, 7, 4, 9).

(a) Find f o g.

(b) Find (f), i.e., the smallest subgroup of S₉ (group of all permutations of 9 elements) which contains the permutation f.

(c) Find f¹⁰⁰ o g¹⁰⁰.

Exercise 4- Suppose we have a field GF(2³) with multiplication modulo x³ + x + 1. Find

(a) all y such that 110(y + 101) = 111,

(b) all y such that y² = 101,

(c) all y suck that y⁸² = 001.