Algebra and number theory, U Glasgow notes by Baker.

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1-2. Use the Euclidean Algorithm and the method of back-substitution to find the following greatest common divisors and in each case express gcd(a, b) as an integer linear combination of a, b: gcd(76, 98), gcd(108, 120), gcd(1008, −520), gcd(936, −876), gcd(−591, 691). Use the tabular method of §4 to check your results. 1-3. 5 agrees with the set T = {t gcd(a, b); t ∈ N0 , 0 < t}. 1-4. 5, show that if n ∈ Z, gcd(12n + 5, 5n + 2) = 1. 1-5. Find all integer solutions x (if there are any) of each of the following congruences: (a) 9x ≡ 23; (b) 21x ≡ 7; (c) 21x ≡ 8; (d) 210x ≡ 97; (e) 13x ≡ 36.

PROBLEM SET 2 45 2-12. Let G = Sym(Tet) be the symmetry group of the regular tetrahedron Tet with vertices A, B, C, D. Let X denote the set of edges of Tet. For ϕ ∈ G and E ∈ X let ϕE = {ϕ(P ) ∈ Tet : P ∈ E}. a) Show that ϕE is an edge and that this defines an action of G on X. b) Find OrbG (E) and StabG (E) for the edge AB. c) For each of the following elements of G find FixG (g): ι, (A B), (A B C), (A B C D), (A B)(C D). 2-13. Let X = {1, 2, 3, 4, 5, 6, 7, 8} and G = (1 2 3 4 5 6)(7 8) be the cyclic subgroup of S7 acting on X in the obvious way.

SUBGROUPS AND LAGRANGE’S THEOREM 37 The Euler ϕ-function ϕ : Z+ −→ N0 is defined by ϕ(n) =number of generators of Z/n =number of elements tn ∈ Z/n with gcd(t, n) = 1. In order to state some properties of ϕ, we need to introduce some notation. For a positive natural number n and a function f defined on the positive natural numbers, the symbol f (d) d|n denotes the sum of all the numbers f (d) where d ranges over all the positive integer divisors of n, including 1 and n. For example, f (d) = f (1) + f (2) + f (3) + f (6).

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