Ring Theory:Euclidean Ring: Prove that a necessary and sufficient condition that the element 'a' in the Euclidean ring is a unit is that d(a) = d(1). Or, Prove that the element 'a' in the Euclidean ring is a unit if and only if d(a) = d(1).

Related BrainMass Solutions

• Ring Theory : Associate of a Greatest Common Divisor and Euclidean Rings

  105221 Associate of a Greatest Common Divisor and Euclidean Rings In a Euclidean ring R, any associate of a greatest common divisor is a greatest common divisor. keywords: GCD, GCDs Please see the attached file for the complete solution.

• Ring Theory: Euclidean Rings and Elements with Least Common Multiples

  105914 Euclidean Rings and Elements with Least Common Multiples Given two elements a, b in the Euclidean ring R their least common multiple cЄR is an element in R such that a│c and b│c and such that whenever a│x and

• Euclidean Rings and GCD

  106672 Ring Theory : Euclidean Rings and GCD If the least common multiple of a and b in the Euclidean ring R is denoted by [a,b], prove that [a,b] = ab/(a, b) where (a, b) is the greatest common divisor of a

• Polynomials over the Rational Field: Euclidean Ring: Let D be a Euclidean ring, F its field of quotients. Prove the Gauss lemma for polynomials with coefficients in D factored as product of polynomials with coefficients in F.

  It is the explanation of the following problem: Let D be a Euclidean ring, F its field of quotients. Prove the Gauss lemma for polynomials with coefficients in D factored as product of polynomials with coefficients in F.

• F[x]/(x^2 + 1) is isomorphic to the field of complex numbers

  For if is a Euclidean ring (or a commutative ring with unit element) and is an ideal of , then is a maximal ideal of if and only if

• Euclidean Rings

  104081 Ring Theory : Euclidean Rings Prove that in a Euclidean ring ( a , b ) can be found as follows b = q0 a + r1 where d ( r1) < d (a ) a = q1 r1 + r2

• F[x]/(f(x)) is a field with p^n elements.

  For if is a Euclidean ring ( or, if is a commutative ring with unit element )and is an ideal of then

• Ring Theory : Prove that the units in a commutative ring with a unit element form an abelian group.

  105744 Ring Theory : commutative ring with a unit element form abelian Prove that the units in a commutative ring with a unit element form an abelian group. Please see the attached file for the complete solution.

• Polynomial Rings: Irreducible Polynomials

  114321 Polynomial Rings: Irreducible Polynomials Prove that x^2 + 1 is irreducible over the field F of integers mod 11 and prove directly that F[x]/(x^2 + 1) is a field having 121 elements.