BSc Mathematics SEM VI 2018 19 May 2018-19 MATHEMATICS PAPER II ALGEBRA Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks for respective subquestions

1. Q1 (a) Answer any ONE
   • i. Let G,G" be groups and f : G be an onto homomorphism. If is a subgroup of G’ then prove that f(h) is a subgroup of G containing ker f. Further show that, if H’ is normal in then f~'(H’) is normal in G 8
   • ii. State and prove the Cayley’s theorem for finite groups. 8
   • (b) Answer any TWO
   • i. Prove that: are cyclic groups and Gj x = {(91, 92) : gi Gi, go with binary operation o defined by 92) = then x is cyclic if and only if and are 6
   • ii. Show the there are two non-isomorphic groups of order 4. 6
   • iii. If G/Z(G) is cyclic then prove that G is an Abelian group. 6
   • iv. Let Qg = {+1, ? = k? = -1 Show that every subgroup of Qs is normal in Qs 6
2. Q2 (a) Answer any ONE
   • i. State and prove the First Isomorphism Theorem(Fundamental theorem of homomorphism) of rings 8
   • ii. Define characteristic of a ring. Show that the characteristic of an in- tegral domain is either zero or a prime. Give example of a ring with characteristic 0 and a ring with characteristic 5 8
   • (b) Answer any TWO
   • i. Define unit and zero divisor in a ring. Show that every element of Z, is either a unit or a zero divisor 6
   • ii. Show that the set of units in a ring R forms a group under multiplication. 6
   • iii. Let be an ideal in a ring R and 7: R > be defined by n(a) =a+T for a R. Show that 7 is a homomorphism and ker 7 = I 6
   • iv. Let S = a,b Z,b is even }. Show that S is a subring of but not an ideal of 6
3. Q3 (a) Answer any ONE
   • i. Show that the only irreducible polynomials in are a linear polyno-. mial x — a or quadratic polynomial 2? + bx + such that b? — 4c < 0, 8
   • ii. Show that an ideal in a commutative ring R is a maximal ideal if and. only if R/M is a field 8
   • (b) Answer any TWO
   • i. Let R be an Integral Domain and p R. Show that if p is prime then p is irreducible. Is the converse true? Justify your answer 6
   • ii. For a commutative ring R, prove that R is a field if and only if {0} is a maximal ideal in R 6
   • iii. Prove that the ring is a field, but is not a field 6
   • iv. Show that a field with characteristic p contains a subfield isomorphic to 6
4. Q4 Answer any THREE
   • (a) Show that for the multiplicative groups R* = R— {0}, R* of 5
   • (b) Let G be a group. Show that the subgroup H = {g? /g G} of G is normal 5
   • (c) Let R be a ring where (R,+) is cyclic, then show that Ris commutative. 5
   • (d) Show that J = are even integers \ is an ideal of 5
   • (e) Find all ideals of Z/12Z using correspondence theorem. 5
   • (f) Show that x” — p is irreducible in for any prime p. 5

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