---- > [!definition] Definition. ([[prime subfield]]) > Since [[characteristic of a ring|characteristic]] is a [[field]]-invariant (see [[field extension]]), the [[category|categories]] $\mathsf{Fld}_{0}$, $\mathsf{Fld}_{p}$ of fields of a given characteristic all live together in $\mathsf{Fld}$ without much interaction. > $\mathbb{Q}$ is [[terminal object|initial]] in $\mathsf{Fld}_{0}$, while $\mathbb{F}_{p}:=\mathbb{Z} / p\mathbb{Z}$ is [[terminal object|initial]] in $\mathsf{Fld}_{p}$. In each case, the initial object is called the **prime subfield** of of the given field. Every [[field]] is a canonical [[field extension|extension]] of one of these fields. > One calls $\mathbb{Q}$ the **prime field of characteristic zero** and $\mathbb{F}_{p}$ the **prime field of characteristic $p$**. ^definition > [!justification] Initiality. > To see $\mathbb{Q}$ is initial in $\mathsf{Fld}_{0}$, let $k$ be any field of [[characteristic of a ring|characteristic]] zero. first recall that since $\mathbb{Z}$ is [[terminal object|initial]] in $\mathsf{Ring}$, we have unique maps $\mathbb{Z}\to \mathbb{Q}$, $\mathbb{Z} \to k$, and these maps are [[injection|injections]] (trivial kernels since we are working over zero characteristic). Any map $\mathbb{Q} \to k$ must 'contain the map $\mathbb{Z} \to k$ inside it'. Recalling that $\mathbb{Q}$ is the [[field of fractions]] of $\mathbb{Z}$, the [[universal property]] of FOF tells us that once we have specified the map $\mathbb{Z} \to k$ its extension to a map $\mathbb{Q} \to k$ exists and is unique. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwBGA4AEUAviDGl0mXPkIoATOSq1GLNgGtJ0kBmx4CRAIyljq+s1aIO3XoOEAtCWNUwoAc3hFQAMwBOEFxIyiA4EEhkalZsnDAAHlhwOHAABACEINQ4dFgMbHwQENpSfoHBiKHhSKbRGjZxiclpmdQMdAIwDAAKsoYKIP5YHnw4WWG5+TaFxTplQZHZEYi1lvW2CUkpGZIUYkA > \begin{tikzcd} > \mathbb{Q} \arrow[rr, "\exists !"] & & k \\ > & \mathbb{Z} \arrow[lu, "\exists !", hook] \arrow[ru, "\exists !"', hook] & > \end{tikzcd} > \end{document} > ``` > > > To see $\mathbb{F}_{p}$ is initial in $\mathsf{Fld}_{p}$, let $k$ be a [[field of fractions|field]] of characteristic $p$. [[first isomorphism theorem for rings|FIT]] gives diagram > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwBGA4AC0AviDGl0mXPkIoAjOSq1GLNgGtJ0kBmx4CRMotX1mrRB269Bw8QHo0nHvyGixAAgC8nl7fdxTwc-dk0YACdPLB0ZA3kiZVNqcw0rThwYAA8cYCwuTzEYsVUYKABzeCJQADMIiC4kMhAcCCQAJila+sbEZtakZTULNk5srDgcOE8AQmjYkDqGjuoBxABmFPVLa3HJ6bnOCBpIhiwwGDyJagY6ARgGAAVZQwUQCKxyvhwFpd7NlptRBDHB0LAMNh8CAQbQlMRAA > \begin{tikzcd} > \mathbb{Z} \arrow[d] \arrow[r, "\exists ! i"] & k \\ > \mathbb{Z}/p\mathbb{Z} = \mathbb{Z} / \ker i \arrow[r, "\exists ! \overline{i}"'] & \text{im }i \arrow[u, hook] > \end{tikzcd} > \end{document} > ``` > > from which existence and uniqueness both follow. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```