A ring is a mathematical structure consisting of a set of elements and two operations, usually addition and multiplication, that follow certain rules. A field is a type of ring where every non-zero element has a multiplicative inverse.
Z[x] is the ring of polynomials with integer coefficients, and pZ[x] is the ideal generated by the polynomial p. Z[x]/pZ[x] is the quotient ring obtained by dividing Z[x] by pZ[x], which essentially means all polynomials in Z[x] are considered equivalent if their difference is divisible by p.
An integral domain is a commutative ring with no zero divisors, meaning that if a and b are non-zero elements, their product ab is also non-zero. In Z[x]/pZ[x], p is a prime number, and since Z[x] is a unique factorization domain, all elements in Z[x]/pZ[x] are either prime or can be factored into primes. This means there are no zero divisors, making it an integral domain.
To prove that Z[x]/pZ[x] is an integral domain, we can use the fact that Z[x] is a unique factorization domain and show that any non-zero element in Z[x]/pZ[x] can be factored into primes. This will demonstrate that there are no zero divisors in the ring, making it an integral domain.
One example is the polynomial x^2+x+1 in Z[x]/2Z[x]. This polynomial cannot be factored into primes, and when multiplied by any other polynomial in Z[x]/2Z[x], the product will never be a multiple of 2, making it a non-zero divisor.