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?

Expert Solution

Colby Messinger answered 2 years ago

Prove that every finite integral domain is a field. Give an example of an integral domain...

Prove that every finite integral domain is a field. Give an example of an integral domain which is not a field. Please show all steps of the proof. Thank you!!

let D be an integral domain. prove that an element of D[x] is a unit if...

let D be an integral domain. prove that an element of D[x] is a unit if an only if it is a unit in D.

Let (Z, N, +, ·) be an ordered integral domain. Let {x1, x2, . . ....

Let (Z, N, +, ·) be an ordered integral domain. Let {x1, x2, . . . , xn} be a subset of Z. Prove there exists an i, 1 ≤ i ≤ n such that xi ≥ xj for all 1 ≤ j ≤ n. Prove that Z is an infinite set. (Remark: How do you tell if a set is infinite??)

Find an example with an explanation of Integral domain but not domain with factorization Domain with...

Find an example with an explanation of Integral domain but not domain with factorization Domain with factorization but not unique factorization domains Domain with factorization but not Noetherian domains Unique factorization domains but not Noetherian unique factorization domains Noetherian Domains but not Noetherian Unique factorization domains Noetherian Unique factorization domains but not principal ideal domain principal ideal domain but not Euclidean domains Euclidean domain but not field

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?

(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)

Complex Variable Evaluate the following integrals: a) int_c (z^2/((z-3i)^2)) dz; c=lzl=5 b) int_c (1/((z^3)(z-4))) dz ;...

Complex Variable Evaluate the following integrals: a) int_c (z^2/((z-3i)^2)) dz; c=lzl=5 b) int_c (1/((z^3)(z-4))) dz ; c= lzl =1 c) int_c (2(z^2)-z+1)/(((z-1)^2)(z+1)) dz ; c= lzl=3 (Details Please)

What is the integral e^z/Z-1 dz where C is the circle|z| = 2?

Find all the six roots of z = 16−16 sqrt (3i).

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

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