= Prime field
{title2=$\mathbb Q$ or $\mathbb F_p$}
The <prime field> of a <field> $K$ is its smallest <subfield>: the <arithmetic> forced upon us by the <identity element> $1_K$. Every <subfield> must contain $1_K$, its repeated sums, their <additive inverses>, and the <multiplicative inverses> of nonzero results. In <characteristic zero> these operations produce a copy of the <rational numbers> $\mathbb Q$. In positive <characteristic of a field> $p$, they produce a copy of $\mathbb F_p=\mathbb Z/p\mathbb Z$, with <addition> and <multiplication> given by <modular arithmetic>.
For example, inside the <field with four elements>, repeated <addition> of $1$ gives $0,1,0,1,\ldots$. Thus its <prime field> is $\{0,1\}$, and the other two elements cannot be reached by repeatedly adding $1$. This is why a <finite field> need not consist entirely of ordinary <integer> residues: a larger <finite field> extends the <arithmetic> of its <prime field> with additional elements.
The positive <characteristic of a field> must be a <prime number>. If its least vanishing multiple of $1_K$ were $rs$ with $1<r,s<rs$, then $(r1_K)(s1_K)=0$ would exhibit <zero divisors>. The absence of <zero divisors> rules this out. The <prime field> supplies the <scalars> when viewing $K$ as a <vector space>; its <basis> then explains the possible sizes of a <finite field>.
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