Kerala University 2009 M.Sc Mathematics Algebra - Question Paper
I (Pages : 3) 4695
Reg. No. : .........................
Name :
Second Semester M.Sc. Degree Examination, August 2009 Branch : Mathematics MM 221 : ALGEBRA (Prior to 2005 Admn.)
Time: 3 Hours Max. Marks: 75
Instructions : 1) Answer 5 questions choosing Part - A or Part - B from each question.
2) All questions carry equal marks.
1. A) a) Prove that Zm x Zn - Zmn if m and n are relatively prime integers. What
can be said about Z2 x Z2 ?
b) Derive the conditions which are necessary and sufficient for a group G to be the internal direct product of its subgroups H and K.
B) a) Show that if m divides the order of a finite abelian group then G has a subgroup of order m.
b) Find, upto isomorphism, all abelian groups of order 60.
c) Show that if G has a composition series and if N is a normal subgroup of G, then G has a composition series.
2. A) a) Let X be a G-set for a group G. Show that Gx = {g e G|xg = x} is a
subgroup of G for each x e X .
b) Show that if X is a G-set for a group G, the relation x1 ~ x2 if and only if x1g = x2 for some g e G, is an equivalence relation on X.
c) Show that every group of order p2 is abelian.
B) a) Show that if H is a p-subgroup of a finite group G and N [H] is the normaliser of H in G, then (N [ H ]: H) = ( G : H) (mod p).
b) Derive the class equation for a finite group G.
c) Show that a group of order 15 has a normal subgroup.
3. A) a) Show that if A is a n x n matrix in F the function ( X, Y ) = XtAY defined
on the space Fn of column vectors is a bilinear form and it is symmetric if and only if A is symmetric.
b) If P is an element in SU2 with eigen values X and X, show that P is
X .
conjugate in SU2 to the matrix
X
c) Show that if A is a skew symmetic matrix, then eA is orthogonal.
B) a) Show that if V is a m-dimensional vector space over a field of characteristic 2 and (,) is a nondegenerate skew-symmetric form on V, then the dimension m of V is even.
b) Show that SU2 is homeomorphic to the unit 3 sphere in IR4.
4. A) a) Show that if E is a finite extension of F and K is a finite extension of E, then K is a finite extension of F and [K:F ]=[K:E][E:F ].
b) Find the degree Q (V2 + V3) over Q.
c) Show that squaring the circle is impossible.
B) a) Let E be a field and F a subfield of E. Show that the set G ( E / F ) of all automorphisms of E leaving F fixed forms a subgroup of the group of all automorphisms of E and F < EG (E/F).
b) If F is a finite field of characteristic p, show that the map op : F F defined by aop = ap is an automorphism of F and FCp ~ Zp.
c) Let E be finite extension of F and o an isomorphism of F onto a field F' and let F7 be an algebraic closure of F. Show that the number of extensions of o to an isomorphism t of E into F7 is finite.
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5. A) a) If E is a field such that F < E < F , show that E is a splitting field over F if and only if every automorphism of F leaving F fixed maps E onto itself.
b) Find the splitting field of X3 - 2 over Q and its degree.
B) a) If E is a finite extension of F and K is a finite extension of E, show that K is a separable extension of F if and only K is a separable extension of E and E is a separable extension of F.
b) If F is a finite field containing q elements and E is a finite extension of degree n over F show that E contains qn elements.
c) Is IR a splitting field of Q ? Is C a splitting field of IR ?
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