---- > [!definition] Definition. ([[field extension]]) > Recalling that every morphism $k \to K$ of [[field|fields]] is an [[injection]], $K$ may be viewed as a particular way to 'enlarge' $k$, since $k$ may be identified with a [[subring|subfield]] of $K$. Notations for such a **field extension** include $k \subset K$ and $K / k$; neither are perfect. > > Note that $K$ is a $k$-[[algebra]] (by the very definition of [[algebra]]), and hence a $k$-[[vector space]] under the induced module structure. ---- #### > [!basicproperties] > Extensions preserve [[characteristic of a ring|characteristic]]. Indeed, $\mathbb{Z}$ is [[terminal object|initial]] in $\mathsf{Ring}$, so there is a unique map $\mathbb{Z} \to K$. Thus the diagram below commutes: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwBGA4AC0AviDGl0mXPkIoyARiq1GLNgGtJ0kBmx4CRJaRXV6zVohABpSaphQA5vCKgAZgCcIXJGRA4EEgmIAx0AjAMAAqyhgognlhOfDg6Ht6+iP6BSABMUuk+wdQ5iLkldFgMbHwQENpiFGJAA > \begin{tikzcd} > \mathbb{Z} \arrow[d] \arrow[rd] & \\ > k \arrow[r, hook] & K > \end{tikzcd} > \end{document} > ``` > Since $k \to K$ is an [[injection]], the [[kernel]] of $i_{k}$ and $i_{K}$ agree. ---- #### 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 > ```