ground fields and ringsGroundFieldsAndRings2006-05-08 19:26:292006-12-31 12:02:54Definitionground fieldbase fieldground ringbase ringground ring\usepackage{latexsym}
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The following is a list of common uses of the \PMlinkescapetext{term} \emph{ground} or \emph{base} field or ring in algebra. These \PMlinkescapetext{terms} are endowed with \PMlinkescapetext{semantics} based on their context so the following list may be \PMlinkescapetext{incomplete} or may not apply uniformly.
One commonality is generally found for the use of ground ring or field: the result is a unitial subring of the original. Outside of this requirement, the constraints are specific to context.
\begin{itemize}
\item Given a ring $R$ with a 1, let $\mathbb{Z}1$ be the subgroup of $R$ generated by $1$ under addition. This is consequently a subring of $R$ of the same characteristic as $R$. Thus is it isomorphic to $\mathbb{Z}/c\mathbb{Z}$ where $c$ is the characteristic of $R$. This is the smallest unital subring of $R$ and so rightfully may be called the ground or base ring of $R$.
When the characteristic of $R$ is prime, $\mathbb{Z}1\cong \mathbb{Z}/p\mathbb{Z}$ and so it may be called the ground field of $R$.
\item Given a vector space or algebra $A$ over a field $k$, then $k$ is the ground/base field of $A$.
\item Given a set of matrices $M_n(R)$, the ground ring is commonly the ring $R$, and if required as a subring of $M_n(R)$ then it is taken as the set of all scalar matrices.
\item Given a field extension $K/k$ over a field $k$, then $k$ is the ground field of $K$ in this context. For a general field where no specific subfield has been specified, the ground/base field then typically defaults to the prime subfield of $K$. (Recall the prime subfield is the unique smallest subfield of $K$.)
\item Given a field $K$ and a set of field automorphisms\, $f:K\rightarrow K$,\, the ground/base field in this context is the \PMlinkname{fixed field}{Fixed} of the automorphisms. That is, the largest subfield of $K$ which is pointwise fixed by each $f$. Since a field automorphism must fix the prime subfield, this definition always produces a field containing the prime subfield.
\end{itemize}