Abstract Algebra Answers

(a) Show that a permutation is even if and only 4 if its signature is + 1. Find the signature of (2 3 4) E S4 using the definition of signature.

(b) Show that, in a finite commutative ring, 6 every non-zero element is either a zero divisor or a unit. Also find the number of zero divisors of Z20.

(a) Show that f : (R 4 , x) -4 (R, +), defined by f (a) = log10a, is an isomorphism of groups, where R+ is the set of positive real numbers.

(b)Give an example of a ring R such that a2 = a for all aER. Show that any such ring is commutative.

(c)Let (C*,.) denote the group of non-zero 3 complex numbers and let S = {zecx I IzI =1}
Show that C*/S~_ R+, where (R+,) is the group of positive real numbers.

(a) Show that any group of order 35 is cyclic.

(b)Use the Eisenstein's criterion for irreduciblity of a polynomial over z [x] to test whether 8x3 + 6x2 — 9x + 24 is irreducible over z [x] or not. Also obtain the quotient field of Q [x]/( 8x3 + 6x2-9x+ 24 )

(c)Let R be a ring and let a E R be such that a2= 1. Let S= fara r E RI. Show that S is a sub ring of R.

(a) Let R and R' be commutative rings and f : R --> R' be a ring homomorphism. Prove or disprove that if I is an ideal of R, then f(I) is an ideal of R'.

(b) Let G be a group and Z (G) ={a E G Ixa=ax for all x E G}. If G/Z (G) is cyclic, then show that G is an Abelian group.

(c) Iet (D. 6) be a Euclidean domain. Prove that for every integer n such that delta (1)+n greater than equal to 0, the function Fn:D\{0}->Z:Fn (a)=6(delta)a+n is a Euclidean valuation on D.

(d) Let G he a cyclic group of order 6. Can G be isomorphic to a subgroup of S7 ? lustily your answer.

(a) Check whether R = Q[x]/<x3+18x2+3x+6> is a field or not. If R is a field, find its characteristic. If R is not a field, obtain its quotient field.

(b) Prove the following stateinent using the principle of mathematical induction :If p (x) ao + a1x + anxn belongs to. R [x] and a belongs to R, then p (x) can be written as bo + b1(x — a) +... +bn (x — a)n, bi belongs to R for all i = 0,1,2,...n.

(c) If x= 2, y = 3,z=4, in Z 5, find .x2 y -2 Z.

(a) Give an example, with justification, of two primes p and q in Z such that p is a prime in Z[i] also but q is not a prime in Z[i].

(b) Let G be a group and a, b E If a5 = e,aba-1= b2, then find the order of b.

(c) Give two distinct elements of , Q[X]/(2X2+7).

1 (i) Let R be a commutative ring with unity and I, J be ideals of R such that I + J - R. Then show
t h a t l intersection J = l J .

(ii) Give an example of a commutative ring R with ideal I and J such that I n J not equal to lJ. Justily your choice of example.


2. (a) Find all the subgroups of the group Z10. Also give the generators of each them.

1. Explain why {2, IGNOU, C t0, 1} is a set.

2. (a) shour rhat ^zsl^l is not a field.
< x r + x + l >

3. let G = {4,8,12,16} proper subset Z20. Make a Cayley table for G for. multiplication. .Use this to check whether G is a group or not.

(c) Find the nil radical of Z5.

Prove that sigma Δ S4, where sigma = 1 2 3 4.