Abstract Algebra Answers

Factorise 10 in two ways in Z[
√
−6]. Hence, show that Z[
√
−6] is not a UFD.

Show that the map f : Z+iZ → Z2, defined by f(a+ib) = (a−b) (mod 2), is an onto ring
homomorphism. Describe kef f. Is it a maximal ideal? Justify your answer.

is it possible for ring of integers to have ideal that is not principal? Justify your answer?

Check whether quartic root of unity {1,-1,i,-i} form ring or not?

a) Show that every group of order 30 has a proper and non trivial normal subgroup.

b) N is a normal subgroup of a group G such that G/N is abelian, then show that [G, G] is a proper subgroup of N.

a) Let R={a+b(root under 5 I a,bEZ} and d:R--->Z be a function defined by d{a +b(root under 5)} = a2 — 5b2.
(i) Check whether d is a ring homomorphism. If it is, find its kernel.

(ii) Show that, if a(alfa) E R is a unit, d{a(alfa)} = -±1.

(b) Check whether {(1, 1), (2, 2), (1, 2), (2, 3), (3, 3), (2, 1), (3, 2)} is an equivalence relation on {1, 2, 3} or not. Give reasons for your answer.

(c) Let Q be the field of rational numbers .
Define a e b= a+ b, a 0 b= 2.b. Check whether (Q, + , .) is a commutative ring with identity.

(a) Find the orders of the elements 7(bar)and 9(bar) in Z12.

(b) Find the signatures of (1 2 4) and (1 3 2 4) in S4, using the definition of signature.

(c) Prove that Q+ Q(root under 3) = {a + b(root under 3) I a, b E Q is a field.

(a) Prove by the method of induction that 12 +22 +32+ ... + n2 n(n+1)(2n+1)/6 for n greater than equal to 1.

(b) Suppose (phie) is a homomorphism from Z30 to Z30 and Ker(phie) = {0, 10, 20}. If (phie)(23) = 9, determine all the elements that image 9 under (phie).

(c) Let G = {[ 1 0] , [ -1 0 ] , [ 1 0 ] , [-1 0 ] }
0 1 0 1 0 1 0 -1
Write down the table of operation where
the operation is matrix multiplication. Is G
a group ? Is G cyclic ? Justify your answer.