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.

Expert Solution

Colby Messinger answered 8 months ago

(8) show that if R is an integral domain with unit element, then any unit in...

(8) show that if R is an integral domain with unit element, then any unit in R[x] is also a unit in R.

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 and D'be integral domains. Let c = charD and c'= charD' (a) Prove that...

Let D and D'be integral domains. Let c = charD and c'= charD' (a) Prove that the direct product D ×D'has unity. (b) Let a ∈D and b∈D'. Prove that (a, b) is a unit in D ×D'⇐⇒ a is a unit in D, b is a unit in D'. (c) Prove that D×D'is never an integral domain. (d) Prove that if c, c'> 0, then char(D ×D') = lcm(c, c') (e) Prove that if c = 0, then char(D...

Let x = inf E. Prove that x is an element of the boundary of E....

Let x = inf E. Prove that x is an element of the boundary of E. Here R is regarded as a metric space with d(p,q) = |q−p| for p,q ∈R

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??)

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic...

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic to a metric space (Y, dY ) of diameter at most 1.

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.

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 G be a cyclic group generated by an element a. a) Prove that if an...

Let G be a cyclic group generated by an element a. a) Prove that if an = e for some n ∈ Z, then G is finite. b) Prove that if G is an infinite cyclic group then it contains no nontrivial finite subgroups. (Hint: use part (a))

Prove that Z[√3i]= a+b√3i : a,b∈Z is an integral domain. What are its units?

Question