PrimeSubfield 2002-05-03 17:16:54 2002-05-03 17:16:54 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\em prime subfield} 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.