Abstract Algebra Answers

Let "a" and "b" be two integers. If "a|b" and "b|a", them show that "a=\\pm b"

Prove that a ring with one consists only of zero if and only if 1 = 0

Show that every group of order 5*7*47 is abelian and cyclic.

Let G be the group of integers under the operation of addition,

and let H = {3k | k ∈ Z}. Is H a subgroup of G

Prove that the set of complex numbers {1,−1,i,−i} under

multiplication operation is a cyclic group.Find the generators of

cycle

3. Let L: R₂ → R₂ be the linear transformation defined by L ([u¹, u₂]) = [u1, 0]

a. Is [0, 2] in Ker L?

b. Is [2,2] in Ker L?

C. Is [3,0] in range L?

d. Is [ 3,2] in range L?

e. Find Ker L.

£. Find Range L.

Prove that Sn is not solvable for n>4.

Let a and b be integers. Prove that if a∣b, then an∣bn for all positive integers n.

Prove or disprove that the polynomial 21x^3 - 3x^2 + 2x + 9 is irreducible over Z2 , but not over Z3. Justify your answer.

Prove or disprove that in Z[x], the ideal <x> + <3> is a principal ideal.