Abstract Algebra Answers

Use the ring Z(underroot -2) to show that (1) the quotient ring of a ufd need not be a UFD. (2)an irreducible element of a UFD need not be a prime element

Which of these is not a set of numbers

a.complex

b.intregral domain

c.real

d.natural

x is a unit in R[x]

1.If (G,*)is a group, then , * is the only binary operation defined on G



2.If every element of a group G has finite order, then G must be of finite order.

The set of cosets of <(1 2)> in S3 is a group with respect to multiplication of cosets.

M3[ Z ]has no nilpotent elements.

Obtain the order of each element of ℘(S) where S={1,2,3}

Let G be a group and H be a non-empty finite subset of .G If ab∈H∀ b,a ∈ H then show that H ≤ .G Will the result remain true if H is not finite? Give reasons for your answer.

Consider the ideal I<x³-1,2x⁴+2x³+7x²+5x+5> in Q[x]. Find p(x) ∈Q[x] such that I =<p(x)> . Is Q[x]/I a field? Give reasons for your answer.

Check whether or not <(x+1)²> is a maximal ideal of Z[x].