Question

In: Advanced Math

(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)

Solutions

Expert Solution

any doubt ask in comment


Related Solutions

Let A = {a+b*sqrt14: a,b∈Z}. Prove that A ∩ Q = Z. Explain is set A...
Let A = {a+b*sqrt14: a,b∈Z}. Prove that A ∩ Q = Z. Explain is set A countable?
Prove that L = {a + b √ 5i | a, b ∈ Q} is a...
Prove that L = {a + b √ 5i | a, b ∈ Q} is a field containing the roots of x2 + 5. Moreover, prove that if Q ⊆ K ⊆ C is a field containing the roots of x2 + 5, then L ⊆ K.
Prove that Z[√3i]= a+b√3i : a,b∈Z is an integral domain. What are its units?
Prove that Z[√3i]= a+b√3i : a,b∈Z is an integral domain. What are its units?
Let A = Z and let a, b ∈ A. Prove if the following binary operations...
Let A = Z and let a, b ∈ A. Prove if the following binary operations are (i) commutative, (2) if they are associative and (3) if they have an identity (if the operations has an identity, give the identity or show that the operation has no identity). (a) (3 points) f(a, b) = a + b + 1 (b) (3 points) f(a, b) = a(b + 1) (c) (3 points) f(a, b) = x2 + xy + y2
let n belongs to N and let a, b belong to Z. prove that a is...
let n belongs to N and let a, b belong to Z. prove that a is congruent to b, mod n, if and only if a and b have the same remainder when divided by n.
Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove...
Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove that xa+b = xaxb b) prove that (xa)-1 = x-a c) establish part a) for arbitrary integers a and b in Z (positive, negative or zero)
b belongs to N. Prove that if {7m, m belong to Z} is not belongs to...
b belongs to N. Prove that if {7m, m belong to Z} is not belongs to {ab, a is integer}, then b =1
(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...
(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2....
(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2. Prove that if 2p−1 is prime, then p must also be prime. (Abstract Algebra)
PROVE THAT COS Z,SIN Z,cosh Z and sinh z are entire function
PROVE THAT COS Z,SIN Z,cosh Z and sinh z are entire function
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT