Question

In: Math

(2) Let Z/nZ be the set of n elements {0, 1, 2, . . . ,...

(2) Let Z/nZ be the set of n elements {0, 1, 2, . . . , n ? 1} with addition and multiplication modulo n. (a) Which element of Z/5Z is the additive identity? Which element is the multiplicative identity? For each nonzero element of Z/5Z, write out its multiplicative inverse. (b) Prove that Z/nZ is a field if and only if n is a prime number. [Hint: first work out why it’s not a field when n isn’t prime. Try some small examples, e.g. n = 4, n = 6.]

Expert Solution

(a) The 0 (zero) element of Z/5Z is the additive identity.

The 1 (one) element of Z/5Z is the additive identity.

(b) If n is not prime, then the multiplicative inverse of each element does not exist. Fore example, let n = 6, then we have

So, we see that if n is not prime then at least (n-1)/2 and even number do not have their inverse. So, it can not be a field.

milcah answered 1 year ago

Let the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider...

Let the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider the following statement: for every positive integer n and every x in Z/nZ, there exists y ∈ Z/nZ such that xy = [1]. (a) Write the negation of this statement. (b) Is the original statement true or false? Justify your answer.

Let A be a set with m elements and B a set of n elements, where...

Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.

Let A be some m*n matrix. Consider the set S = {z : Az = 0}....

Let A be some m*n matrix. Consider the set S = {z : Az = 0}. First show that this is a vector space. Now show that n = p+q where p = rank(A) and q = dim(S). Here is how to do it. Let the vectors x1, . . . , xp be such that Ax1, . . . ,Axp form a basis of the column space of A (thus each x can be chosen to be some unit...

Prove (Z/mZ)/(nZ/mZ) is isomorphic to Z/nZ where n and m are integers greater than 1 and...

Prove (Z/mZ)/(nZ/mZ) is isomorphic to Z/nZ where n and m are integers greater than 1 and n divides m.

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove...

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove (a). (a^m)(a^n)=a^(m+n) (b). (a^m)^n=a^(mn)

7. Let n ∈ N with n > 1 and let P be the set of...

7. Let n ∈ N with n > 1 and let P be the set of polynomials with coefficients in R. (a) We define a relation, T, on P as follows: Let f, g ∈ P. Then we say f T g if f −g = c for some c ∈ R. Show that T is an equivalence relation on P. (b) Let R be the set of equivalence classes of P and let F : R → P be...

Let f(n,k) be the number of equivalence relations with k classes on set with n elements....

Let f(n,k) be the number of equivalence relations with k classes on set with n elements. a) What is f(2,4)? b) what is f(4,2)? c) Give a combinational proof that f(n,k) = f(n-1,k-1)+k * f(n-1,k)

Let Z ∼ Normal(0 , 1) and Y ∼ χ 2 γ , then the new...

Let Z ∼ Normal(0 , 1) and Y ∼ χ 2 γ , then the new r.v. T = √ Z Y /γ has the Student’s t-distribution. The density function of T is fT (t) = Γ[(γ + 1)/2] √γπΓ(γ/2) 1 + t 2 γ !−(γ+1)/2 . (a) (3 points) Describe the similarity/difference between T and Z. (b) (6 points) Let t0 be a particular value of t. Use t-distribution table to find t0 values such that the following statements...

Let Dn be the set of positive integers that divide evenly into n. List the elements...

Let Dn be the set of positive integers that divide evenly into n. List the elements of each of the sets D6, D16, D12, and D30

Let A = [ 0 2 0 1 0 2 0 1 0 ]  . (a)...

Let A = [ 0 2 0 1 0 2 0 1 0 ]  . (a) Find the eigenvalues of A and bases of the corresponding eigenspaces. (b) Which of the eigenspaces is a line through the origin? Write down two vectors parallel to this line. (c) Find a plane W ⊂ R 3 such that for any w ∈ W one has Aw ∈ W , or explain why such a plain does not exist. (d) Write down explicitly...