Definition 17.
A field is said to be prime if it is isomorphic to \(\Q\) or to one of the finite fields \(\F_p\text{.}\)
Section Prime subfield
We begin with the definition of a prime subfield of a field \(F\text{.}\)
Theorem 18.
Let \(R\) be a ring with a subfield. Then \(R\) has a unique subfield \(P\) which is a prime field. Furthermore, \(P\) is contained in every subfield of \(R\text{,}\) and also in the center of \(R\text{.}\)
Proof.
Let \(f\colon\Z\to R\) be a unique ring homomorphism, and let \(E\subseteq R\) be a subfield of \(R\text{.}\) We put \(K=E\cap Z(R)\text{.}\) Note that \(f(\Z)\subseteq K\text{.}\) Therefore, \(\ker f\) is either \(0\) or it is generated by some prime number \(p\text{.}\) If \(\ker f=\{0\}\text{,}\) then \(f\) is injective and \(f\) can be extended to a field isomomorphism of \(\Q\) with a subfield \(P\subseteq K\text{.}\) If \(\ker f=(p)\) then, \(f\) defines an isomorphism from \(\F_p\) to a subfield \(P'\subseteq K\text{.}\)
