(a)Show that S = {a+b √ 5 | a, b ∈ Q} is a subring of...

(a)Show that S = {a+b √ 5 | a, b ∈ Q} is a subring of the real numbers (with the usual + and × of real numbers). Explain why S is a field.

(b) Prove that if r is an element of a ring R and r 3 = 0, then 1 − r is a unit in R.

(c) Write down all the nilpotent elements of Z24, stating the index of nilpotence in each case. Verify the statement in part (b) holds in Z24.

Expert Solution

Colby Messinger answered 2 years ago

Show that {a + b√3 | a, b ∈ Q} is a field (a subfield of...

Show that {a + b√3 | a, b ∈ Q} is a field (a subfield of the field R), but {a + b√3 | a, b ∈ Z} is not a field.

(a) Prove that Q(sqareroot 5)={a+b sqareroot 5 ; a,b in Z} is a subring of Z....

(a) Prove that Q(sqareroot 5)={a+b sqareroot 5 ; a,b in Z} is a subring of Z. (b) Show that Q(sqareroot 5) is a conmutative ring. (c) Show that Q(sqareroot 5) has a multiplicative identity. (d) show that Q(sqareroot 5) is a field.(Hint : you want to mulitply something by he conjugate.) (Abstract Algebra)

Use rules of inference to show that the hypotheses p → q, r → s, and...

Use rules of inference to show that the hypotheses p → q, r → s, and ¬q ∨ ¬s implies ¬p ∨ ¬r

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S = Z (integers), R = {(a,b) : a = b mod 5}. Is this relation...

S = Z (integers), R = {(a,b) : a = b mod 5}. Is this relation an equivalence relation on S? S = Z (integers), R = {(a,b) : a = b mod 3}. Is this relation an equivalence relation on S? If so, what are the equivalence classes?

Suppose S = {p, q, r, s, t, u} and A = {p, q, s, t}...

Suppose S = {p, q, r, s, t, u} and A = {p, q, s, t} and B = {r, s, t, u} are events. x p q r s t u p(x) 0.15 0.25 0.2 0.15 0.1 (a) Determine what must be p(s). (b) Find p(A), p(B) and p(A∩B). (c) Determine whether A and B are independent. Explain. (d) Arer A and B mutually exclusive? Explain. (e) Does this table represent a probability istribution of any random variable? Explain.

The supply function for oil is given (in dollars) by S(q), and D(q). S(q)=q^2+15q, D(q)=1024-17q-q^2 a....

The supply function for oil is given (in dollars) by S(q), and D(q). S(q)=q^2+15q, D(q)=1024-17q-q^2 a. Graph the supply and demand curves. b. Find the point at which supply and demand are in equilibrium. c. Find the consumers' surplus. d. Find the producers' surplus.

On the set S of all real numbers, define a relation R = {(a, b):a ≤ b}. Show that R is transitive.

please show all the steps and in order Q.5 Let X and Y be continuous rvs...

please show all the steps and in order Q.5 Let X and Y be continuous rvs with the joint pdf f(x, y) = 60(x^2)y, for 0 < x, 0 < y, 0 < x + y < 1 and 0 otherwise. (a) Find E[X + Y ] and E[X − Y ] (b) Find E[XY ] (c) Find E[Y |X = x] and E[X|Y = y]. (d) Find Cov[X, Y ]

Q1: The expression A = (E – F)/((P + (3Q))((2R) – (5S))) is not a valid...

Q1: The expression A = (E – F)/((P + (3*Q))*((2*R) – (5*S))) is not a valid SPLan assignment statement. Write a sequence of assignment statements that will perform the equivalent calculation. Q2: Write a program which asks the user to input an integer. If the integer is less than zero, the program should output the message “Negative, the absolute value is ”, followed by the absolute value of the number. If the input is zero, the program should output the...

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