Kannur University 2011-1st Sem M.Sc Mathematics s ( Syllabus) MODEL MAT1C01: Algebra-I - Question Paper

PART A
1. State the fundamental theorem of finitely generated abelian groups. obtain all abelian
groups, up to isomorphism, of order 1089
2. describe the normalizer of a subgroup of a group. Let G be a finite group and let p be a
prime dividingG. Let P be a Sylow p-subgroup ofG. Show thatN[N[P]]?N[P].
3. describe a free abelian group. Show that ? under addition is not a free abelian group.
4. Let F be a field and let f(x)?F[x]be a non constant polynomial. When do you say
that f(x)is irreducible overF? Show that x4?2x2?8x?1is irreducible over ?.
5. describe prime and maximal ideals of a ring. Prove that if Fis a field, every proper
nontrivial prime ideal of F[x]is a maximal ideal.
PART B
6. (a)If m divides the order of a finite abelian groupG, then show that G has a subgroup of
order m.
(b)If G is indecomposable, show that G is cyclic with order a power of a prime.
7. State and prove the 1st isomorphism theorem.
8. describe a free abelian group. Let G?{0}be a free abelian group with a finite basis. Show
that any 2 finite bases of G have the identical number of elements.
9. Let H be subgroup of a group G and N be normal subgroup of G. Show that ? ?????
and ???? ????? ???.
10. (a)Prove that if D is an integral domain, then D[x]is an integral domain and the units of
D[x]are precisely the units ofD.
(b) obtain a polynomial of degree?0 in ????? that is a unit.
11. Let H be a subring of the ring R. Show that the multiplication of additive cosets of H is
well described by the formula (a?H)(b?H)?ab?Hif and only if ah?Hand hb?H
for alla,b?Randh?H.
PART C
UNIT 1
12. (a) Let ? be a ????? and ? ??. Show that if ??? is finite, and then???? is a divisor of ???
(b) Let G be a group of order pnwhere p is prime and let X be a finite G-set. Show
that (mod ) G X?X p
13(a) If ? and ? are distinct primes with ???, then show that every group Gof order ??
has a single subgroup of order ? and this subgroup is normal inG.
(b) Prove that no group of order 160 is simple.
UNIT II
14. describe a subnormal series of a groupG. Prove that 2 subnormal series of a group G
have isomorphic refinements.
15. (a) Let Fbe a field of quotients of Dand let Lbe any field containingD. Show that
there exists a map ?:F? Lthat provide an isomorphism of Fwith a subfield of Lsuch
that ?(a)?afora?D.
(b) Show that any 2 fields of quotients of an integral domain Dare isomorphic.
UNIT III
16. (a) State and establish Eisenstein criterion for irreducibility of a polynomial in ????G
(b) Show that the polynomial () 1
1
p
p
x x
x
?
? ?
?
is irreducible over ? for any primep.
(c) Show that for pa prime, the polynomial xp?ain ?????is not irreducible for any
? ???.
17. (a) If F is a field, show that every ideal in F[x]is principal.
(b) An idealp(x)?{0}ofF[x]is maximal if and only if p(x)is irreducible overF.

Earning:   Approval pending.