Give an example, with justification, to show that:
i) the union of integral domains need not be an integral domain;
ii) all finite fields are not isomorphic.
i) The union of integral domains is not necessarily an integral domain. Consider the integral domains Z (the set of all integers) and Z[i] (the set of complex numbers of the form a + bi, where a and b are integers and i is the imaginary unit). Both Z and Z[i] are integral domains, as they satisfy all the properties required for integral domains. However, their union, which includes both real and complex numbers, is not an integral domain. For example, the element 1
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Consider the ring homomorphism 
. Show that Kerφ = (x,3) . What does the Fundamental Theorem of Homomorphism say in this case?
Which of the following statements are true? Give reasons for your answers. Marks will only be given for valid reasons.
i) If σ is an even permutation, then σ2 = 1.
ii) If G is a group such that o(G) = 2m , where m∈N , then G has a subgroup of order m.
iii) If G = < X > is of order 25, then xα generates G , where α is a factor of 25.
iv) {α1, α2,....αn} is a set only if all the s iα follow a given rule.
v) The characteristic of a field containing (50 − 1) elements is 50.
vi) Every subring of a non-commutative ring is non-commutative.
vii) If (R, ∗1,∗2) is a ring, then ψ : R X R→ R : ψ(r1,r2) = r1 ∗2 r1 is a binary operation.
viii) Not every polynomial that is irreducible over Q[x] is irreducible over Z[x] .
ix) If (D, +,.) is an integral domain such that 1∈ D , then D is a field.
x) The function f , defined by f(x) = x - 1/3 - x , has the same set as domain and as range.
Prove that if G ≠ {e} and G has no proper non-trivial subgroup, then G is finite and O(G) is a prime number.
Check whether or not {Z∈C|Z5 = 1 is a group with respect to addition.
Give an example, with justification, to show that:
i) the union of integral domains need not be an integral domain;
ii) all finite fields are not isomorphic.
Let p be a prime and a ∈ N such that p|a50 . Show that p50|a50 .
For x∈G , define Hx = {g-1Xg | g ∈G}. Under what conditions on x will Hx ≤ G ? Further, if Hx ≤ G , will Hx _∆ G ? Give reasons for your answer.