Reference no: EM131205731
Exercise 1- Find all generators and all subgroups of Zx29. 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.
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