State the definition of an integral domain State the definit

State the definition of an integral domain. State the definition of the greatest common divisor in a polynomial ring. State the definition of an irreducible polynomial.

Solution

Integral domain:Let R be a commutative ring with unit(1).Then R is called an integral domain if r has no zero divisors.

Greatest Common Divisor of a Polynomial ring:let X be a non emty subset of a commutative ring R.An element d of R is a greatest common divisor of X provided:

1.d divides a for all a belongs to X

2.c divides a for all a belongs to X i.e c divides d

Irreducible polynomial:Let R be an integral domain with unity.A polynomial f(x) belongs to R[x] is said to be irreducible over R,if whenever f(x)=g(x)h(x),where g(x),h(x) both belong to R[x],then either deg(g(x))=0 or deg(h(x))=0 i.e either g(x) or h(x) is a constant polynomial.