Integral domains

• Why Is Every Finite Integral Domain a Field?
  In a finite integral domain, multiplication by any nonzero element is injective and therefore surjective, which guarantees a multiplicative inverse.
• Does Having No Roots Make a Degree-2 or Degree-3 Polynomial Irreducible?
  For polynomials of degree 2 or 3, rootlessness implies irreducibility over a field, but not over every integral domain; learn the needed conditions.