Abstract Algebra Answers

Prove that an element of an integral domain is a unit iff it generates the domain.

Show that d:QQ[x]\{0}→NN∪ {0}:d(f)=2^(deg f) is a Euclidean valuation on QQ[x].

Let G be a group, H⍙G and β ≤ G/H . Let A = {x ∈G | Hx∈β} . Show that i) A ≤ G , ii) H⍙A , iii) β = A/H.

Show that Zto Z/nZ is a natural epimorphism

Find unit set of Z√m. Where m is not perfect square.

Let p be a prime and a ∈N such that 50 a|p . Show that 50 50 p .

Which of the following statements are true? Give reasons for your answers. Marks will
only be given for valid reasons.
i) If σ is an even permutation, then I
2 σ = .
ii) If G is a group such that m (o G) = 2 , where m∈N , then G has a subgroup of order
m.
iii) If G is of order 25, then = < x >
α
x generates G , where α is a factor of 25.
iv) { , , , } α1 α2 K αn
is a set only if all the s αi
follow a given rule.
v) The characteristic of a field containing )1 50( − elements is 50.
vi) Every subring of a non-commutative ring is non-commutative.
vii) If ) ( ,R ,
1 2
∗ ∗ is a ring, then 1 2 1 2 1 ψ : R × R → R :ψ(r r, ) = r ∗ r is a binary operation.
viii) Not every polynomial that is irreducible over ]x[ Q is irreducible over ]x[ Z .
ix) If .), ( ,D + is an integral domain such that 1∈D , then D is a field.
x) The function f , defined by
3 x
x 1
)x(f
−
−
= , has the same set as domain and as
range.

Prove that R^(n)/R^(m) ~ R^(n-m)
as groups, where n, m∈ N, n ≥ m.

Prove that if G ≠ {e} and G has no proper non-trivial subgroup, then G is finite and o(G) is a prime number.

How many Sylow 5-subgroups, Sylow 3-subgroups and Sylow 2-subgroups can a
group of order 200 have? Give reasons for your answers.