ABSTRACT
In the previous chapter we proved that any irreducible polynomial f over a field F has a unique splitting field: a minimal field extension of F in which f can be factored into linear factors. This provides a field inside of which we can explore whether or not the roots of f are obtainable by elementary algebraic operations. We will pursue this goal in the remaining chapters in this book. But a wonderful bonus flows from the existence and uniqueness of
splitting fields, and we will take a small detour from our goal to explore this bonus in the present chapter. We are now able to completely describe all finite fields. We have for a long time been familiar with the finite fields Zp, where p is a prime integer. We have also encountered various finite fields as finite extensions of such fields; for example, consider Example 42.5, Exercise 43.2 and Example 45.5. In this chapter we will be able to place these examples in a beautiful general context.
Theorem 42.2 says that every field has characteristic zero or p, where p is a prime integer. Fields with characteristic zero have a subfield isomorphic to Q and so are infinite. Thus, any finite field has characteristic p, for some prime p. This means that every finite field contains (an isomorphic copy of) one of the fields Zp as a subfield (recall that this is called the prime subfield). We use these considerations to prove our first result about arbitrary finite fields.
