Unique Factorization Domain with Quotient Field

Related BrainMass Solutions

• If R is a unique factorization domain and if a and b in R are relatively prime (i.e.,(a,b) = 1), whenever a divides bc, then a divides c. That is, if R is a unique factorization domain and if a and b in R are relatively prime (i.e., (a,b) = 1), whenever a divides bc then a divides c.

  132134 If R is a unique factorization domain and if a and b in R are relatively prime (i.e.,(a,b) = 1), whenever a divides bc, then a divides c.

• If R is a unique factorization domain and if a,b are in R, then a and b have a least common multiple (l.c.m.) in R.

  132136 If R is a unique factorization domain and if a,b are in R, then a and b have a least common multiple (l.c.m.) in R.

• unique factorization domain

  Since R is an unique factorization domain (UFD), then we have unique factorizations for both a and b a = p_1 * p_2 * ... * p_k b = q_1 * q_2 * ... * q_t where p_1, ..., p_k are prime factors of a, q_1, ..., q_t are prime factors of b Then ab = p1

• R-module proof

  If R is a principal ideal domain.... Please see the attachment. Proof: Because is a principal ideal domain, then is also a unique factorization domain. For any nontrivial ideal in , we have is generated by some element .

• Functions

  The solution provides examples of working with synthetic division, rational zero theorem, domain, asymptotes.

• Maximal Ideals, Cosets, Polynomials and Quotient Rings

  Is the Z[x] / (2, ) an integral domain (You need to prove it) Please see the attached file for the complete solution. Thanks for using BrainMass. Maximal Ideals, Cosets, Polynomials and Quotient Rings are investigated.

• Square roots of matrix

  One way exploits the eigenvector-value factorization A = Q Q ^T =Q  Q^TQ Q^T.

• division algorithm

  To show that M is maximal, let's show that the quotient ring is a field.

• Simplification by Factorization and finding domain

  182004 Simplification by Factorization and finding domain Problem 1) Simplify 5xy-15x/4xy Problem 2) x2 -9x +18 = 0 (x-6)(x-3) = 0 x =6 x =3 What happens when x is 6 or 3. What is the domain if x can only be 6 or 3? 1.