# prime subfield

The of a field $F$ is the intersection of all subfields of $F$, or equivalently the smallest subfield of $F$. It can also be constructed by taking the quotient field of the additive subgroup of $F$ generated by the multiplicative identity $1$.

If $F$ has characteristic $p$ where $p>0$ is a prime, then the prime subfield of $F$ is isomorphic to the field $\mathbb{Z}/p\mathbb{Z}$ of integers mod $p$. When $F$ has characteristic zero, the prime subfield of $F$ is isomorphic to the field $\mathbb{Q}$ of rational numbers.

Title prime subfield PrimeSubfield 2013-03-22 12:37:47 2013-03-22 12:37:47 djao (24) djao (24) 4 djao (24) Definition msc 12E99