Let z be an element of P and factor z in the ufd. Let f be one of the factors, a polynomial that is both prime and irreducible. Since P is minimal, f spans all of P. In other words, a hypersurface is defined by an irreducible polynomial. For example, a paraboloid is a surface in 3 space defined by a quadratic polynomial such as z = x2+y2.
Conversely, let f be an irreducible polynomial, generating a prime ideal P. If P contains a smaller, minimal prime ideal than f generates an irreducible polynomial, which is a contradiction. Thus P is a minimal nonzero prime ideal. By the previous theorem, P is part of a chain of length n+1, with 0 below and n-1 primes above. Therefore the dimension of P′ is n-1, producing a hyper surface. The irreducible polynomials define the hyper surfaces in Cn.
If the generating polynomial is linear the hypersurface is a hyperplane.
The prefix hyper indicates we are working in n dimensional space, and the surface has n-1 dimensions. However, this is pretty clear by context, so we sometimes drop the prefix and talk about surfaces and planes in Cn. By assumption, these are n-1 dimensional structures. In 3 space they become the surfaces and planes that you know and love.
Suppose f and g are distinct irreducible polynomials that produce the same surface. They live in the same prime ideal, hence they generate each other. This makes f and g associates in the polynomial ring. The units in this ring are the nonzero elements of K. Thus f is a scale multiple of g. Divide through by the lead coefficient and make f monic. This is the canonical polynomial for its surface. In other words, surfaces and monic irreducible polynomials correspond 1 for 1.