These are 7 parts of question 2. Kindly answer them with proper reason and examp

Question

These are 7 parts of question 2. Kindly answer them with proper reason and example thanks.

Question 2: Answer the following questions: (a) Is every set which has addition and multiplication defined on it a field? If it is, explain why. If not, give a counter-example (an example of a set with operations of addition and multiplication which is not a field) (b) Is it possible to define vector spaces over fields other than the set of real numbers R? If it is, give an example of one such vector space. (e) What does it mean when it is said that the set Ze, for a positive integer e, partitions the integers? (d) For Ze, where c is a positive integer, if we want to convert this set into a field under modular addition and multiplication, why can we pick a unique representative from each equivalence class to work with? (Why can we choose x ? to represent the whole set in terms of modular addition and multiplication)? (e) Is the set Ze, where c is any positive integer, together with modular addition and multiplication a field for alc? In which cases is Ze with the previously defined operations a field? (f) Given any field F, what is one way you can construct a vector space of dimension n over F? (g) Based on the problems you worked on for part 2 of this project, what are the advantages of working with vectors spaces defined over finite fields instead of working over vector spaces defined over R?

Explanation / Answer

(a)The set of integers Z has addition and multiplication defined on it.but it is not a field since no non zero element except 1 and -1 has a multiplicative inverse in Z.

(b)Yes,R over Q,the set of rational numbers is a field.

(c)It means that Zc gives rise to an equivalene relation defined by a~b iff a-b is divisible by c.under this relation Z is partioned into c equivalence classes whose union is again the whole of Z.

(d)We can work with the representative because any element of the class differs from the representative by a multiple of c.when we work with addition or multiplication this difference will give zero.

(e)No,Zc is not a field for all c.e.g.Z6 is not a field,since 3 does not have a multiplucative inverse here.

Zc is a field iff c is a prime number.

(f)F×F×.......×F (n times) is a n dimensional vector space over F.

(g)the advantage of working with finite dimensional vector spaces over finite field is that all the sets are finite.

Thanks

Happy chegging!!!

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