Prime (sub)fields

Overview and definition

As a consequence of the homomorphism induced on the field of quotients on any field $F$, we can identify the unique minimal subfield of $F$ called the prime subfield, which is generated by its multiplicative identity $1\in F$. Every field $F$ has a unique subfield isomorphic to $\mathbb{Q}$ or ${\mathbb{F}}_p$, depending on its characteristic.

Prime (sub)field

Let $F$ be a field and $f:\mathbb{Z}\to F$ be the unique homomorphism defined by $f(n)=n\cdot 1$.

• If $F$ has characteristic $0$, then $f$ is injective and induces an injective homomorphism $\tilde{f}:\mathbb{Q}\to F$. In particular, the image of $f$ the smallest subfield of $F$ (hence unique) and isomorphic to $\mathbb{Q}$.
• If $F$ has positive characteristic $p$, then $f$ has kernel $⟨p⟩$ and hence image isomorphic to $\mathbb{Z}/p\mathbb{Z}={\mathbb{F}}_p$, the ring of integers modulo prime $p$.

The unique subfields isomorphic to $\mathbb{Q}$ and ${\mathbb{F}}_p$ for fields of characteristic $0$ and $p$, respectively, are called prime subfields. The fields $\mathbb{Q}$ and ${\mathbb{F}}_p$ are themselves called prime fields and contain no proper subfields.

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Connections to other topics

• Extension fields: Every field is an extension of its prime subfield.
• Finite extension fields: Every finite field (which has characteristic prime $p>0$) arises as a simple extension of its prime subfield $\mathbb{F}={\mathbb{F}}_p(\gamma )$, where $\gamma$ is a generator of the cyclic group $({\mathbb{F}}^{\ast },\cdot )$.
• Galois groups: If $E$ is a finite extension of its prime subfield $F_0$, then every automorphism $\sigma :E\to E$ necessarily fixes $F_0$, since by definition $\sigma (1)=1$ fixes the generator $1\in \mathbb{Q}$ if $\text{char}(E)=0$, and preserves addition and multiplication mod $p$ if $\text{char}(E)=p$. Hence the Galois group of $E$ as an extension of $F_0$ is precisely $\text{Gal}(E/F_0)=\text{Aut}E,$ the set of all automorphisms $E\to E$.