Exercise 10.2. Let D be an integral domain, regarding as a subring of Frac(D) (the field of quotients of D). Show that for every ring R with D ≤ R ≤ Frac(D), we have Frac(R) ≈ Frac(D).

Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero in

Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]

18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .

Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.

Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.

15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .

Let Q be the field of rational numbers, then show that Q(√2, √3)= Q(√2 + √3)

a) The ring R[x, y]/(x + 1) is a field.b) The ring Z[x]/(7) is a principal ideal domain.   please if able explain each taken step in detail, I'm quite new to abstract algebra, thank you in advance.

Determine if R is (1) a field (2) an integral domain (3) a unital ring, where R= {2+y√P+2√z,y,z e Q. p.q prime).