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
______ ______ __________ _________ _______ _________.
______ _________ _____ ________ ______ __________ _________ ______ __________ __________ _________ _________.
________ _____ _______ _________ ____ _______.
____ ________ _____ _________ ___ ____.
__________ ______ ____ ___ ___ __________ ________.
_____ ________ ____ __________ ___ ________ ______ __________ ____ _____ _______.
____ ____ __________ ________ _____ ________ __________ ______ ________ ______.
__________ _________ _________ _______ ________ _______ _____ ______ _____ __________ __________.
______ _____ ___ _______ ______ ___ ___ __________ __________ ______ _____ __________.
_________ _____ _______ __________ _______ _______ ___ _______.
____ __________ ________ ________ ___ ____ _____.
____ ____ __________ __________ ________.
_______ ______ ____ ________ ________ ________.
_____ ___ _____ _________ ________.
__________ ____ _____ _______ ________ ________ ______ __________ ____ ____.
_____ ___ _____ _________ _______ ______ ________ __________.
_______ __________ __________ _________ _____ ____ ____ __________ __________ __________ __________ __________.
_____ ______ _____ _________ ____ ___ ________ ___.
___ ________ _________ ____.
- Solved Assignments
- Study Notes
- Guess Papers
- Question Papers
✓ Unlock up to 60% of the answer
✓ No payment required
Let p be a prime. Show that S = {m + sqrt(-p)|m, n∈Z} is a subring of C. Further, check whether or not S is an ideal of C.
Consider the ring homomorphism 
. Show that Kerφ = (x,3) . What does the Fundamental Theorem of Homomorphism say in this case?
Check whether f :(4Z, +) → ( Z4, +) : f (4m) = is a group homomorphism or not. If it is, what does the Fundamental Theorem of Homomorphism give us in this case? If f is not a homomorphism, obtain the range of f .
Let G be a group, H∆G and H β≤G/H . Let A = {x∈ G|Hx ∈ β}. Show that
i) A ≤ G , ii) H ∆ A , iii) β = A/H .
Give an example, with justification, of each of the following:
i) a zero divisor in Z5 ,
ii) an element of C[0, 1] which is not a zero divisor,
iii) a subring of an integral domain which is not an integral domain.
Write the Cayley tables for addition and multiplication in Z7 .
Show that f d : Q[X]\ {0} → N∪{0} :d(f) = 2degf is a Euclidean valuation on Q[x] .