Abstract Algebra Answers

Prove that field has no zero divisors

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 .

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.