Max-spectrum of polynomial ring over a field
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Let be a field. The max-spectrum of the polynomial ring
can be described as follows.
As a set
Here are some descriptions:
- It is the set of all irreducible monic polynomials in
. A particularly distinguished class of irreducible polynomials is the monic linear polynomials, which can be identified with the points of
itself. When
is algebraically closed, these are the only elements.
- It is the set
. Here
denotes the algebraic closure of
and
is the group of automorphisms. The notation here means that we take points of
, and identify those which have the same minimal polynomial. Note that distinct elements of
remain distinct, and when
itself is algebraically closed, these are the only elements.
Note that the max-spectrum of the polynomial ring over any field is always infinite, whether or not the field is algebraically closed and whether or not the field is finite. This is because there are always infinitely many irreducible polynomials.
As a topological space
The max-spec topology is the cofinite topology: the proper closed subsets are precisely the finite ones. Thus, the space is a T1 space and is also irreducible and Noetherian.