Find K^G where 

Let G be a group of automorphisms of a field K. Is the fixed field  K^G a subfield of K ? Why, or why not?

Use the division algorithm to find the inverse of .

Check whether or not  is a Euclidean domain

Check whether 9782957210008 is a valid ISBN number.

Evaluate the following Legendre Symbols:

Give two distinct rings whose quotient field is {a ib a, b }. + Justify your answer.

Check whether or not the ring 

i) is finite;
ii) has zero divisors;
iii) has nilpotent elements.

Construct the free group on the set Further, check if it is a free abelian group or not.

Find all the possible abelian groups, up to isomorphism, of order 900.

Show that SU(2) and S³ are structurally the same.

Write as a product of O(3) and an element of B₃ ( ).

Let G1 and G2 be finite groups such that p divides  | G₁ | and  | G₂ | . Prove that the Sylow p-subgroups of G₁ G₂ are precisely of the form P₁ P₂ ,  where P₁ and P₂ are Sylow p-subgroups of G₁ and G₂ respectively.

Find the number of Sylow 5-subgroups, Sylow 7-subgroups and Sylow 2-subgroups A₅ has.

Let G be a finite group and H be a normal subgroup of G . Prove that  where the C_x are all the distinct conjugacy classes of G such that 

Let G be the group of all rigid motions of a plane and S be the set of all rectangles in the plane. Show that G acts on S . Also obtain the orbit and stabiliser of a square under this action.

Let G be a group and let H G, K G, o(H) o(K) p, a prime. Show that either H K ={e}  = or H =K. = Is this result still true if p is not a prime? Justify your answer.

Which of the following statements are true? Give reasons for your answers. Marks will only be given for valid justification of your answers.

(i) If G is a finite abelian group and p is a prime factor of o(G), then the number of Sylow p-subgroups of G is a prime.

ii) The minimum polynomial of 1/3 5 over Q is  x^1/3 .

iii) 

iv) If G is a finite group and m o(G),  then G has a subgroup of order m.

v) If  and  are two 15^th roots of unity, then 

vi) There exists an extension field of of order 25.

vii) Every group of order 18 has a normal subgroup of order 2.

viii) If I and J are ideals of a ring R, then IJ = 

ix) If f : R S → is a ring homomorphism and I is an ideal of R, then f (I) is an ideal of S.

x) Every prime ideal of an integral domain is a maximal ideal.

Set up geometric data tables for a unit cube using only i) vertex and polygon tables and ii) a single polygon table. Compare the two methods for representing the unit cube with a representation using three data tables, and estimate storage requirements for each.

Modify the boundary-fill algorithm for a 4-connected region to avoid excessive stacking by incorporating scan line methods.