Which general methods of field construction do we know?

Question

This question is partially motivated by: https://mathoverflow.net/questions/3003/in-what-sense-are-fields-an-algebraic-theory whereas I am not a specialist in field theory.

Q1: Which methods do we know to construct fields? Is there a complete catalog of methods of field construction?

The linked question states that there is not a notion such as "free field generated by a set", as in contrast to, say, "free group generated by a set". But a fairly general method to construct fields is the quotient field of an integral domain. Which other methods do we know, and do they yield different fields?

Q2: Which methods do we know if we require the field to be ordered or even Archimedian?

Answer 1

Here are some common field constructions:

• Fields of fractions of integral domains.

• Quotient rings $K[X]/(p)$, where $K$ is a field and $p$ is an irreducible polynomial over $K$.

• Taking the algebraic closure of a field.

• Taking the intersection of subfields of a field.

• Taking the composite of subfields of a field.

Answer 2

In a sense we know how all fields can be constructed from "a few" small ones: in general every field $K$ contains a field $P$ which is either isomorphic to $\mathbb{Q}$ or to a finite field $\mathbb{Z}/p\mathbb{Z}$.

The field extension $K/P$ possesses a transcendence basis $T$, so that $K$ is an algebraic extension of the field of fractions $K(T)$ of the polynomial ring $K[T]$ in the set of variables $t\in T$. Even more precisely this means that $K/P$ can be split into the following steps: an algebraic extension $P^\prime /P$, the extension $P^\prime (T)/P^\prime$, an algebraic extension $K/P^\prime(T)$.

The structure of the first kind of extensions is known, if $P$ is finite, otherwise it is the subject of algebraic number theory and at least "a lot is known".

So after all taking fraction fields of polynomial rings and taking factor rings modulo an irreducible is all that is needed to built all fields.