Page "Zariski topology" Paragraph 38

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.

If k is not algebraically closed, for example the field of real numbers, the picture becomes more complicated because of the existence of non-linear irreducible polynomials.

For example, the spectrum of &# 8477 ; consists of closed points ( x − a ), for a in &# 8477 ;, ( x < sup > 2 </ sup > + px + q ) where p, q are in &# 8477 ; and with negative discriminant p < sup > 2 </ sup > − 4q < 0, and finally a generic point ( 0 ).

For any field, the closed subsets of Spec k are finite unions of closed points, and the whole space.

( This is clear from the above discussion for algebraically closed fields.

The proof of the general case requires some commutative algebra, namely the fact, that the Krull dimension of k is one — see Krull's principal ideal theorem ).