(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 1 year 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)

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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.

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

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Let R be the relation on Q defined by a/b R c/d iff ad=bc. Show that R is an equivalence relation. Describe the elements of the equivalence class of 2/3.

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